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1999 Quarter 3
Term Structure Economics
from A to B
2
by Joseph G. Haubrich
Depositor-Preference Laws
and the Cost of Debt Capital
10
by William P. Osterberg and James B. Thomson
Household Production
and Development
21
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by Stephen L. Parente, Richard Rogerson, and Randall Wright
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FEDERAL RESERVE BANK
OF CLEVELAND
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Vol. 35, No. 3
http://clevelandfed.org/research/review/
Economic Review 1999 Q3
1999 Quarter 3
Term Structure Economics
from A to B
2
by Joseph G. Haubrich
Depositor-Preference Laws
and the Cost of Debt Capital
10
by William P. Osterberg and James B. Thomson
Household Production
and Development
by Stephen L. Parente, Richard Rogerson, and Randall Wright
21
1
ECONOMIC REVIEW
1999 Quarter 3
Vol. 35, No. 3
Term Structure Economics
from A to B
2
by Joseph G. Haubrich
The interest rates for bonds of different maturities are related, but the
interplay of factors that influence these rates is not easy to tease apart.
The author leads the reader through the development of a model of the term
structure of interest rates, then works with the model to provide some insights
into the interplay of factors, especially the effect of uncertainty on interest
rates. His analysis shows how a common simplification known as the
expectations hypothesis obscures the significant contribution that uncertainty
can make to the determination of interest rates.
Depositor-Preference Laws
and the Cost of Debt Capital
10
by William P. Osterberg and James B. Thomson
Under depositor-preference laws, depositors’ claims on the assets of failed
depository institutions are senior to unsecured general-creditor claims.
As a result, depositor preference changes the capital structure of banks
and thrifts, thereby affecting the cost of capital for depositories. Depositor
preference has no impact on the total value of banks and thrifts, however,
unless deposit insurance is mispriced.
Household Production
and Development
21
by Stephen L. Parente, Richard Rogerson,
and Randall Wright
The authors introduce home production into the neoclassical growth
model and examine its consequences for development economics,
focusing on how differences in policies that distort capital accumulation
explain international income differences. In models with home production, such policies not only reduce capital accumulation, they also
change the mix of market and nonmarket activity; therefore, for a given
policy differential, these models generate larger differences in output
than standard models. Policy differences’ (hence market income
differences’) welfare implications change when the model explicitly
incorporates home production.
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Editors: Monica Crabtree-Reusser
Michele Lachman
Deborah Zorska
Design: Michael Galka
Typography: MAC Services
Opinions stated in Economic
Review are those of the authors
and not necessarily those of the
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Material may be reprinted
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ISSN 0013-0281
2
Term Structure Economics
from A to B
by Joseph G. Haubrich
Joseph G. Haubrich is a consultant
and economist at the Federal
Reserve Bank of Cleveland. He
thanks George Pennacchi for
helpful comments.
Introduction
Most people understand that the term “interest
rates” is plural and acknowledge the difference
between the rates on a savings account, overnight federal funds rate, and 10-year Treasury
bonds. Of the many differences one can point
to, such as risk, issuer, or denomination, among
the most basic and most important factors for
determining the interest rate is the maturity, or
length, of the bond. In this case, a surprisingly
small amount of economics can yield some
valuable insights into the relationship between
interest rates on bonds of various maturities, or
what is more often called the term structure of
interest rates.
Economics tells us that at the most basic
level, interest rates are a price that borrowers
pay investors for moving purchasing power
from the present to the future. This price obviously has both real and nominal components
—the future value of the money you invest will
depend on how high inflation is in the meantime. The price also reflects aspects of risk.
Because you’re uncertain exactly what you’ll
need for retirement, you’re uncertain about
how much consumption you should transfer
into the future. Real variables, inflation, and
uncertainty interact in rather complex ways,
and some common perspectives ignore factors
that play a key role in determining interest
rates. A careful look with an economist’s eye
can sort out these different effects.
Term Structure versus Yield Curve
Two closely related but distinct terms are often
used interchangeably. If we are interested in how
interest rates vary with maturity, it is useful to
look at the yield curve, which plots the yield to
maturity of different bonds against maturity. The
problem is that most Treasury bonds are coupon
bonds, paying a fixed amount semi-annually. For
the purist then, the yield on a five-year T-bond
is really an average of the five-year interest rate
on the principal and many shorter rates on the
coupon payments. One solution is to look at
yields on zero-coupon bonds, which have no
coupons. Figure 1 shows the recent yield curves
for coupon and zero-coupon bonds. Some liquidity and tax differences between coupons and
zeroes lead many to prefer to estimate the pure
interest rates, known as the term structure (of
interest rates) from coupon bonds. See McCulloch,
Huston, and Kwon (1993) and Dhillon and Lasser
(1998) for a discussion of this. So, while the term
structure is the more useful theoretical concept,
the yield curve is easier to observe.
3
F I G U R E
Yield Curve for October 5, 1999
I. Real Term
Structure
1
a
Yield
6.6
6.4
Zero-coupon b
6.2
6.0
Coupon
5.8
5.6
5.4
5.2
5.0
4.8
0
5
10
15
20
25
30
Years to maturity
a. All instruments are Treasury constant-maturity series.
b. For each maturity, the yield is the average of yields on zero-coupon Treasury bonds with that maturity, as of October 5, 1999.
SOURCE: Wall Street Journal, October 5, 1999, p. H15.
F I G U R E
2
Endowment, Preferences, and
Interest Rates
Consumption
tomorrow
Indifference curve
Endowment
point
Interest rate is slope
of budget line
To understand the interplay of factors that
determine interest rates, it is easier to begin by
ignoring the problem of inflation and think of
real bonds. Given that a dollar tomorrow will
buy just as much beef, beer, or baby-sitting
time as a dollar will today, we further simplify
and talk about bonds in terms of abstract
consumption units (although, for the sake of
concreteness, it sometimes helps to think of it
as ice cream).
The economic logic behind interest rates
represents an application of supply and
demand. The interest rate serves as the price
expressing the trade-off of consuming today
versus consuming tomorrow. It adjusts to
equate the supply of savings with the demand
for savings. Even at this general level, we can
note that an increase in the demand for savings
will increase interest rates. If we specialize further, we can answer more specific questions,
such as how recessions or economic growth
affect interest rates.
The first step is to aggregate everyone in the
economy into a single representative agent and
to consider the choice problem of this agent.1
The second step is to consider an endowment
economy without production. The consumption
good just drops from the trees. The last step is
to assume no storage possibilities. In other
words, bonds are in “zero net supply,” so that
when someone is borrowing, someone is lending. Any individual can save or borrow by
using a “consumption loan,” say, giving up one
unit of consumption today for some units
tomorrow, but the economy as a whole cannot.
Thus, in a very simple two-period case, in
equilibrium the interest rate will adjust to make
the representative agent content to hold her
endowment. In figure 2, this is seen as the line
tangent to the agent’s indifference curve at the
endowment point. The basic idea behind this
simple case—where preferences and the
amount of consumption today and consumption tomorrow determine the interest rate—
extends to more complicated cases with
uncertainty and many time periods.
Consumption today
SOURCE: Author’s calculations.
■ 1 Of course, different investors may have different preferences.
Wang (1996) considers this case.
4
Many Periods: A
More Formal Approach
Extending this analysis to many periods and to
uncertainty about future consumption requires a
more formal, mathematical approach. This section
sets up such a model.
There is a single representative agent with
preferences
¥
(1) E 0 ^ t=0
b t u (ct ),
where E 0 denotes expectations as of period 0,
b denotes the discount factor, and u(ct ) denotes the utility of consumption in period t.
The agent faces a budget constraint,
(2) ct +B1t +B2t £ dt +B1t –1 R1t –1+B2t –2 R2t–2,
where Bjt ,j =1,2 is the amount of a bond of
length j bought in period t. These bonds are
perfectly safe, and at the beginning of period t
investors know the gross rates of return R1t and
R2t . The endowment, or dividend, for a period
is denoted dt .
The agent seeks to arrange consumption
to maximize utility, subject to the budget constraint, so a natural way to solve the problem
is to substitute (2) into (1) and obtain the firstorder conditions.2
J =E 0 ^ ¥t =0 b t u(dt +B1t –1 R1t –1+B2t –2 R2t–2 –B1t –B2t ),
¶J
t
¶B1t=0=E 0[–b u' (dt +B1t –1 R1t –1+B2t –2 R2t –2 –B1t –B2t )
+ b t+1 R1t u' (dt+1+B1t R1t+B2t R2t –B1t –B2t )]
and
¶J
t
¶B2t=0=E 0[–b u' (dt +B1t –1 R 1t –1+B2t –2 R 2t–2 –B1t –B2t )
+ b t+2R2t u'(dt+2+B1t+1R1t+1+B2t +1R2t +1–B1t +1–B2t +1)].
We can simplify this in two ways. First, we use
(2) again to get consumption back into the
equations. Next, we take the perspective of
time period t, where R1t , R2t , and ct are
known, which allows us to drop some of
the expectations operators. We get
(3) u' (ct ) = bEt [R 1t u' (ct+1 ) ]
and
(4) u' (ct ) = b 2Et [R2t u' (ct+2) ] .
These have an intuitive explanation. The
left-hand side is the marginal utility of consuming one unit less in period t, that is,
what you give up (in utility terms) if you
invest. The right-hand side tells you what
you gain: the discounted expected marginal
utility of an extra R1t units of consumption
in a future period. The agent equates marginal cost and marginal benefits, leading to
equations (3) and (4).
To focus on the interest rates, it is useful to
rearrange (3) and (4) as
(5)
u' (ct+1 )
1
]
= bEt [
u' (ct )
R 1t
and
(6)
u' (ct+2 )
1
].
= b 2E t [
u' (ct )
R2t
The left-hand sides of (5) and (6) are the
current (date t ) prices of a bond that pays one
unit of consumption one period or two periods
in the future. The lower the price, that is, the
less you pay for such a bond, the higher the
interest rate: Bond prices and rates move in
opposite directions.
Even here we are not quite finished. Both
R1t and R2t are gross returns, and R2t in particular is a two-period gross return. For example,
if the interest rate is 10 percent, R1t is 1.10 and
R2t is 1.21. Because we want to compare the
returns on bonds of different maturities, however, we need to standardize the returns—if
one period is a year, we would want to annualize the returns. To transform R2t into a oneperiod return we can take the square root.3 The
annualized return on the long (that is, twoperiod) bond is then
L t== R2t .
This simplified model, expressed by
equations (5) and (6), is the basis of an
analysis that can give us a lot of insight into
the term structure.
■ 2 Although the budget constraint assumes that the representative
agent holds only one- and two-period bonds, the equilibrium interest rates
on these bonds will be the same even if the agent can hold bonds of other
maturities.
■ 3 This makes sense in the discrete time framework. In some cases, it
is more convenient to take logarithms. See Campbell, Lo, and MacKinlay
(1997), chapter 1.
5
The Expectations
Hypothesis and
Beyond
What do equations (5) and (6) tell us about the
term structure? A good place to start is the simple case of no uncertainty, where the consumer
knows everything today—all future interest
rates and all future consumption endowments.
Then we can rewrite (6) as
(7)
u' (ct +1 ) u' (ct+2 )
1
1 .
1
2
R 2t= b [ u' (ct ) u' (c t+1)] = R1t • R1t+1
Put differently, Lt== R 1t R 1t+1. The long-term
rate is the average between today’s short-term
rate and tomorrow’s short-term rate. The rational investor has two ways of moving consumption from t into t +2: invest in a long bond
with per-period return L t , or roll over a shortterm bond, getting rate R 1t at the start and R 1t+1
next period. The two ways of investing must
have the same return; otherwise, the investor
moves her savings from the low-return investment to the high-return investment. So, in the
case of perfect certainty, the interest rates of
long- and short-term bonds will adjust to keep
today’s long rate an average of today’s and
tomorrow’s short rate.
Equation (7) is often seen in a slightly
modified form, which, although not exactly
correct, is often useful when high precision is
not necessary. This approximation to (7)
takes the form
Lt – 1 =
(R1t –1) + (R1t+1 –1) .
2
For example, if interest rates are 3 percent
today and 7 percent tomorrow, the long-term
rate should be 5 percent. This is not quite
exact, as Lt = = (1.03)(1.07) = 1.0498; but, for
many purposes, it is close enough.
Perfect certainty, as anyone who watches the
stock market can attest, is a rather unrealistic
assumption. One common way to incorporate
uncertainty is to replace unknown future rates
in (7) by their expectation. Thus,
(7a) Lt = = R 1t Et R 1t +1 .
The long-term rate is an average of current and
expected future short-term rates. This is often
■ 4 The extent to which the expectations hypothesis is a good
approximation to the data is a much discussed issue. See Campbell, Lo,
and MacKinlay (1997), section 10.2.
termed the expectations hypothesis of the term
structure. A useful approach, it is not derived
from (5) and (6), and it ignores the risk effect
of uncertain interest rates.4
A more correct treatment with uncertainty
comes from a closer look at equations (5) and
(6). Rewrite (6) as
(8)
u' (ct+1 ) u' (ct +2 )
1
b
]
= Et [ b
u' (ct )
u' (ct +1)
R2t
or
(9)
u' (ct +1 )
u' (ct +2 )
1
E b
]
=E [b
u' (ct ) t+1 u' (ct +1 )
R 2t t
or
(10)
u' (ct +1 ) 1 .
1
]
=E [b
u' (ct ) R1t +1
R2t t
To split out the risk terms, we use the standard
formula
(11)
E (XY) = E (X) E (Y) + cov(X,Y),
where cov stands for the covariance of X and
Y. Using (11), (10) becomes
(12)
u' (ct +1 )
1
1
= Et [ b
]E t [
]
R 2t
u' (ct )
R 1t +1
+ cov [ b
u' (ct +1)
u' (ct +2 )
,b
].
u' (ct )
u' (ct +1 )
A little more work yields
u'(ct +1 ) u'(ct +2)
1 1
1
(13)
= Et [
]+cov[ b
,b
].
R2t R1t R1t+1
u'(ct ) u'(ct +1 )
A lot of insight about the effect of uncertainty comes from comparing (13), the correct
model with uncertainty, with (7), the correct
model with perfect certainty, and (7a), the
expectations hypothesis. The correct interest
rate differs from the simple expectations
hypothesis in two ways. The first is a Jensen’s
inequality term that would arise even with riskneutral investors. The second way is a risk premium that arises precisely because investors are
not risk neutral.
6
The Jensen’s inequality term arises because
Et [ 1 ] does not equal 1 ; indeed,
E t R 1t+1
R1t+1
1
1
Et [
] ³
. The reason is that a
R 1t+1
Et R 1t+1
given change in the interest rate has more
effect on the price of a bond when rates are
low than when rates are high.5 The difference
can be large. Hearkening back to the simple
numerical example above, suppose short-term
rates stand at 3 percent today and are expected
to be 7 percent tomorrow, but have an even
chance of being either at 3 percent or 11 percent. Using the Jensen’s inequality part of (13)
(ignoring the covariance term) the (annualized)
yield on the long-term bond is
Lt =1/= (1.03)/[0.5 1 + 0.5 1 ] = 1.0487.
1.03
1.11
Thus, correctly considering uncertainty leads
to an interest rate of 4.87 percent— a bit below
the 4.98 percent suggested by the simple
expectations hypothesis. You might not notice
this on your savings account, but if you were a
pension fund investing millions of dollars, it
would add up.
This example highlights another key feature
of the model: The interest rate can change significantly even if expected rates stay constant. If
future rates become more or less uncertain,
rates will change today. The numerical example
showed this quite clearly: If future short-term
interest rates were known with certainty to
be 5 percent, then the long-term rate would be
4.98 percent. When those future rates became
uncertain, the long-term rate fell to 4.87 percent.
The second way the model in (13) differs
from the expectations hypothesis is that interest
rates also have a risk premium. In focussing on
the Jensen inequality term, we’ve ignored the
covariance terms —in a sense, we’ve said that
the world got riskier, but nobody cared. And
we’ve also ignored the underlying link with
consumption—when the whole point of the
exercise is to stop taking interest rates as given
and consider their underlying determinants.
Casual inspection of (13) suggests that this general investigation might get quite complicated,
as we have a covariance term involving nonlinear functions of consumption in three time periods. In this case, discretion is the better part of
valor, and it makes sense to examine some simplified versions of the general problem.
■ 5 For an excellent discussion of this point, see Litterman,
Scheinkman, and Weiss (1991).
■ 6 For a more general version of this approach, see Campbell
(1986), and Campbell, Lo, and MacKinlay (1997), chapter 11. See also
Sargent (1987), section 3.5.
A Specialized
Example
By making a number of special assumptions,
we can get to a series of explicit equations that
make it easy to look at the effects of various
factors on the term structure. First, specialize to
log utility,6 so that u(c) = log(c).
Recalling that in equilibrium, consumption
must equal the dividend endowment for the
representative agent, equations (5) and (6)
reduce to
d
1
= bEt ( t )
d
R 1t
t+1
and
d
1
= b2Et ( t ) .
dt+2
R2t
We further specialize by specifying a particular
stochastic process for the dividends: We base it
on an AR(1) process, of the form log dt +1 = g +
r log dt + Ut+1, where Ut+1 is a sequence of
independent and identically distributed random
variables. Adding a time trend (and normalizing
the growth rate g, the process for dividends is
given by:
¥
(14) log dt+1 = gt + ^k=0
r k Ut-k .
We further assume that the Ut+1 terms are
distributed log-normally. This lets us invoke the
useful substitution that if X is distributed lognormally,
logE (X) =E [log(X) ] + 12–VAR [log(X)].
It also helps if we change the definition of
interest rate slightly. We have been thinking
about rates on a discrete time basis; if the
yearly interest rate R1t is 1.05, an investment
of $1 returns $1.05 at the end of the year,
and it is natural to say that the interest rate is
5 percent. When we start using logs, however,
it is more convenient to consider continuously
compounded rates of return, leading to the
definitions
rt = log R1t
and
lt = log Lt = log = R2t .
7
The difference between the two definitions is
often small: log(1.05)=0.0488.
Taking (14) as the dividend process, these
various assumptions allow equations (5) and
(6) to take a relatively convenient, if not exactly
simple, form:
(15) r1t = log 1 +g +(r –1) ^¥k=0 r k Ut-k – 12– sU2
b
and
¥
(16) lt =log 1 +g+ 21– (r 2 –1)^k=0
r k Ut-k – –41 (1+ r 2 ) sU2 .
b
What Moves the
Term Structure?
Equations (15) and (16) provide a reference
point for illustrating term-structure economics.
A variety of factors will move interest rates and
the term structure. These include the value of
today’s shock Ut , the persistence of the endowment shocks r, the growth trend g, the variance
of the endowment shocks sU2 , and the time
preference parameter b. Of these, the most
interesting are Ut , g, and sU2 .
How do interest rates react to the endowment shock Ut ? Simple calculus shows that
]r1t ( r –1 )
=
]Ut
and
]l t 1 ( r 2 –1) .
= 2–
]Ut
A positive shock today lowers interest rates
as long as r ,1. Income today is relatively high,
so people want to save the extra income; consequently, they drive up the price of bonds and
correspondingly drive down the interest rate. In
addition, the term structure steepens because
short rates fall more than long rates. This is
because with r ,1, the effect of the shock dies
off, so that if income is high today, it is also
expected to be higher than average next period,
but not quite so high. The size of the effect,
and thus the incentive to save, diminishes,
leading to a smaller increase in long rates.
A somewhat different picture emerges if
r .1. Then, an increase today means an even
bigger increase tomorrow, depressing the
incentive to save and increasing rates.7 If r =1,
then the shock has no effect on interest rates—
income is expected to go up exactly as much in
the next period, so there is no change in the
demand for saving.
This intuition follows through to the case of
changes to the growth rate of endowments, g.
]r
]l
In that case, ]g1t =1, and ]gt =1. Growing
dividends means that future dividends are
expected to be greater than current dividends
(similar to the case for r .1). An increase in the
growth rate means that future dividends will be
increasingly greater than current dividends,
leading to a lessening of the desire to save
today. This lower demand for savings, and thus
for bonds, decreases bond prices and increases
interest rates. An increase in the growth rate of
dividends increases both short- and long-term
interest rates one for one.
Changing the stochastic process of the
dividends will also change the term structure.
Consider the effects of an increase in the variance of the shocks to income, s U2 . In this case,
]r1t
= – 1–2
]sU2
and
]l t
= – 4–1 (1+ r 2 ).
]sU2
The increased uncertainty lowers both shortand long-term rates. The basic intuition is that
as uncertainty increases, investors wish to save
more “for a rainy day.” The increased demand
for saving drives down interest rates.8 The yield
curve steepens as long as r ,1, because if
shocks die out, an increased variance is less
important the further out it is, and the demand
for savings responds correspondingly less.
Notice that though an increase in uncertainty
leads to a steeper term structure, this happens
not because long rates rise, but because they
do not fall as far as short rates. In some sense,
the increase in uncertainty is proportionally not
so bad for the long term as for the short term,
and thus has less of an impact on long-bond
prices. This result must be interpreted carefully,
however, because with a log-normal distribution, changing the variance of shocks also
■ 7 With r ,1, some delicate issues arise about the existence of
solutions to equation (1). For a discussion, see Campbell (1986) or
Labadie (1994).
■ 8 Not every utility function displays such behavior, so the result is
not completely general. See Zeldes (1989) for a good discussion.
8
changes the mean of the distribution. Increasing
the variance here does not induce a meanpreserving spread.
II. Nominal Term
Structure
In the real world, the vast majority of bonds
pay off in dollars — not in gold, sides of
beef, or Internet-connect time. Some bonds are
indexed for inflation, but, in the United States at
least, most are not. This means that bonds do
not have a certain payoff in consumption
terms—you don’t know for sure what $1,000
will be worth in 10 years—and bond pricing
must take inflation risk into account.9
Fortunately, the analysis of section I can
accommodate the shift to nominal interest rates
relatively easily. Start by considering the nominal return on a bond, R 1t$ , and note that to convert the dollars into consumption units and get
a real return, we must consider inflation )t +1•10
The nominal return on a bond is constant, so
we can get a revised version of equation (5) as
(17)
1 = bE [ u' (ct +1 ) 1 ]
t
$
R 1t
u'(ct ) ) t +1
or
(18)
$=
R 1t
=
1
u' (ct +1) 1
bE t [
]
u'(ct ) )t+1
1
.
u'
(c
)
1 E 1 +b cov(
t+1
, 1)
u' (ct ) )t+1
R1t ()t+1)
Equation (18) has a classic simplification due
to Irving Fisher. Note that if consumption and
inflation are perfectly certain (that is, there is
no uncertainty), (18) reduces to R $1t = R1t•)t +1.
Shifting the perspective to rates, R $1t –1'R1t –
1+)t +1. That is, a nominal interest rate of 5 percent may be broken into a real interest rate of
3 percent and an inflation rate of 2 percent.
Notice that even with perfect certainty, this
approximation does not hold for high interest
rates: While 5 percent is a good approximation
to (1.03)(1.02)=1.0506, 50 percent is not such a
good approximation to (1.30)(1.20)=1.56.
■ 9 For several approaches to adding inflation to a term structure
model, see Sun (1992), Campbell, Lo, and MacKinlay (1997), section
11.2.1, Labadie (1994), and den Haan (1995). Sargent (1987) provides an
in-depth view of monetary economies.
With uncertainty, the simplification becomes
an even worse approximation. As illustrated
earlier, uncertainty has two components. One is
the Jensen’s inequality term. The other is the
risk premium, the covariance between the real
interest rate (or consumption) and inflation.
Notice that this term can be positive or negative. Uncertainty about inflation may move
interest rates up or down. This may seem counterintuitive, but it makes sense. For example, if
inflation covaries positively with consumption
growth, a nominal bond acts as a sort of insurance. If we get lucky next period, and have a
high income, we regret having saved a lot —
but a high inflation rate reduces the value of
our savings. If we are unlucky, and income is
low next period, we wish we had saved
more—but a low inflation rate increases the
value of our savings. Positive covariance,
though, is probably not the most important
case. A variety of studies find that inflation is
negatively correlated with consumption growth
(or, equivalently, real interest rates; see Pennacchi
[1991]), so that inflation risk in fact increases
interest rates. The risk premium is positive.
Looking at longer rates merely compounds
the effect of uncertainty. Thus we have
(19)
1 = b 2E [ u' (ct +1) 1 u' (ct +2) 1 ] .
t
$
R 2t
u'(ct ) )t +1 u'(ct +1) )t +2
Longer-term rates depend on how the real
economy will evolve, how the price level will
move, and the interactions between the two.
Of course, simplifying assumptions can make
(19) easier to interpret, but its more challenging
form is probably more useful. What will higher
consumption growth do to interest rates? Trick
question—we don’t really know until we have
decided what will happen to inflation, and how
inflation will react to the higher growth.
III. Conclusion
The Roman poet Horace once remarked that
getting rid of folly was the beginning of wisdom. Something similar might be said of the
term structure. Understanding the simplifications involved in averaging current and future
interest rates or in subtracting off expected
■ 10 More precisely, let P $1t be the dollar price of a pure discount
bond in time t with one period left to maturity. That is, the bond will pay $1
in period t +1. Let the price level ($/unit of consumption good) be Q t . Then
the real return is R 1$t = 1$ • Q t .
P 1t Q t +1
9
References
Returns and Compounding
This article mostly uses the simple net return as a measure
Pt +1
–1. Exactly what
Pt
this rate is depends on the length of the period, although
in the financial press, returns are usually annualized and
expressed as if the return were for one year. Academic
work often uses continuously compounded returns,
P
rt = log ( t +1), because they simplify calculations.
Pt
Using continuously compounded rates, equation (7)
becomes
u' (ct +1) u' (ct +2)
e –2lt = b 2 [
] = e –r 1t • e –r 1t +1.
u' (ct ) u' (ct )
of the interest rate, defined as rt =
This implies that lt = 12 (r1t +r1t +1), making the long rate
an exact average of the current and future short rates.
Similarly, using continuously compounded rates for (18)
would give an exact Fisher equation, r 1t$ =r1t + pt +1.
inflation is one benefit of looking at the deeper
theory of interest rates. Another benefit arises
from a better understanding of how uncertainty
influences interest rates.
By suggesting that current long-term interest
rates are an average of current and expected
short-term rates, the expectations hypothesis
captures an important truth. But it is not the
whole truth. We have seen how changes in the
uncertainty surrounding future rates may
change the term structure, even if expected
rates stay the same.
The effects of uncertainty are more varied,
and often more subtle, than many people
realize. An increased uncertainty about future
interest rates has an effect on the Jensen’s
inequality factor that tends to lower long-term
interest rates today. An increased uncertainty
about future consumption has an effect on the
risk premium that tends to lower interest rates
today, as people save for a rainy day, but it
steepens the term structure. An increased
uncertainty about inflation will increase nominal interest rates, at least if inflation and consumption covary negatively. The effects on
longer rates are more complicated.
The real world is undoubtedly more complex
than the model of interest rates considered
here. Like a map, which can never show every
detail, our model can highlight important and
dangerous areas of that rather mysterious area
known as the term structure. This can lead to
better decisions, be they on the part of particular investors or of monetary policymakers.
Examining the underlying economic theory
becomes the first step in understanding the
interplay between real and nominal risk factors,
where they come from, and how changes in
those factors matter.
Campbell, John Y. “Bond and Stock Returns
in a Simple Exchange Model,” The Quarterly
Journal of Economics, vol. 101, no. 4
(November 1986), pp. 785– 803.
Campbell, John Y., Andrew W. Lo, and
A. Craig MacKinlay. The Econometrics of
Financial Markets. Princeton, N. J.: Princeton
University Press, 1997.
den Haan, Wouter J. “The Term Structure of
Interest Rates in Real and Monetary
Economies,” Journal of Economic Dynamics
and Control, vol. 19, no. 5/7 (1995),
pp. 909–40.
Dhillon, Upinder S., and Dennis J. Lasser.
“Term Premium Estimates from ZeroCoupon Bonds: New Evidence on the
Expectations Hypothesis,” The Journal of
Fixed Income, vol. 8, no. 1 ( June 1998),
pp. 52–58.
Labadie, Pamela. “The Term Structure of
Interest Rates over the Business Cycle,”
Journal of Economic Dynamics and Control,
vol. 18, no. 3/4 (May–July 1994), pp. 671– 97.
Litterman, Robert, José Scheinkman, and
Laurence Weiss. “Volatility and the Yield
Curve,” The Journal of Fixed Income, vol. 1,
no. 1 ( June 1991), pp. 49 –53.
McCulloch, J. Huston, and Heon-Chul Kwon.
“U.S. Term Structure Data, 1947–91,” Ohio
State University, Working Paper no. 93 – 6
(March 1993).
Pennacchi, George. “Identifying the Dynamics
of Real Interest Rates and Inflation: Evidence
Using Survey Data,” Review of Financial
Studies, vol. 4, no. 1 (1991) pp. 53– 86.
Sargent, Thomas J., Dynamic Macroeconomic
Theory. Cambridge, Mass.: Harvard University
Press, 1987.
Sun, Tong-sheng. “Real and Nominal Interest
Rates: A Discrete-Time Model and Its
Continuous-Time Limit,” Review of Financial
Studies, vol. 5, no. 4 (1992), pp. 581– 611.
Wang, Jiang. “The Term Structure of Interest
Rates in a Pure Exchange Economy with
Heterogeneous Investors,” Journal of
Financial Economics, vol. 41, no. 1 (1996),
pp. 75–110.
Zeldes, Stephen P. “Optimal Consumption
with Stochastic Income: Deviations from
Certainty Equivalence,” Quarterly Journal
of Economics, vol. 104, no. 2 (May 1989),
pp. 275–98.
10
Depositor-Preference Laws
and the Cost of Debt Capital
by William P. Osterberg and James B. Thomson
William P. Osterberg is a senior
economist and James B. Thomson
is a vice president and economist
at the Federal Reserve Bank of
Cleveland.
Introduction
The subsidy inherent in the current depositinsurance system creates perverse incentives
for risk taking by insured depository institutions
(Kane [1985]). The thrift debacle and its attendant financial and political costs have exposed
the dangers of combining virtually unlimited
federal deposit guarantees and regulatory
discretion. Federal deposit guarantees, the toobig-to-let-fail doctrine, and capital forbearance
programs have effectively limited markets’ ability
to discipline troubled institutions. On the other
hand, principal-agent conflicts have often
produced government regulatory policies
designed to forestall disciplinary actions
against troubled banks and thrifts (Kane
[1989]; Thomson [1992]).
Armed with increased awareness of the role
played by regulatory forbearance in the thrift
debacle and in record losses from the 1980s’
bank closings, Congress passed the Federal
Deposit Insurance Corporation Improvement
Act of 1991.1 The FDICIA contains four important reforms: First, it requires prompt corrective
action for undercapitalized banks and for those
considered problem institutions by their primary federal regulator.2 Second, the FDICIA
limits Federal Reserve discount-window loans
to troubled depositories.3 Third, the FDICIA
now requires the Federal Deposit Insurance
Corporation to charge insured institutions a
risk-related deposit-insurance premium. Finally,
the FDICIA replaces the too-big-to-let-fail doctrine with the systemic-risk exception, which
codifies the terms and conditions under which
the FDIC can bail out uninsured claimants of
failed depositories.4
■ 1 DeGennaro and Thomson (1996) show that capital forbearance
increased the total taxpayer bill in the thrift debacle more than 500 percent.
■ 2 Carnell (1993) notes that the FDICIA does not remove regulatory
discretion but progressively limits it as an institution slides toward
insolvency.
■ 3 Todd (1993) argues that these discount-window provisions are
designed to prevent the Federal Reserve from propping up insolvent banks
through improper solvency-based loans.
■ 4 Carnell (1993) contends that abuse of the exception can be
limited by FDICIA provisions requiring written authorization from the
Federal Reserve Chairman and the Secretary of the Treasury for financing
systemic-risk losses by a special assessment on banks’ total liabilities
(total deposits).
11
Shortly after enacting the FDICIA, Congress
added another potentially important measure to
limit the FDIC’s (and hence taxpayers’) exposure. The Omnibus Budget Reconciliation Act
of 1993 created a national depositor-preference
law, changing the priority of depositors’ (and
thus the FDIC’s) claims on the assets of failed
banks by making other senior claimants subordinate to depositors.5 In other words, Congress
implemented depositor preference in an effort
to reduce the FDIC’s losses by changing the
capital structure of banks.
This paper analyzes the impact of depositorpreference laws on banks’ cost of debt capital
and on the value of FDIC deposit guarantees.6
We extend the single-period-cash-flow version
of the capital-asset pricing model, presented by
Chen (1978) and modified by Osterberg and
Thomson (1990, 1991), to include depositor
preference. In this model, the value of a firm is
the present value of its future cash flows. The
values of a firm’s debt and equity are the present values of these claims on the firm’s cash
flows. Riskless cash flows are discounted at the
risk-free rate of interest. Risky cash flows are
converted to certainty-equivalent cash flows by
deducting a risk premium from the expected
cash flow. In this model, the risk premium is
simply the market price of risk, multiplied by
the covariance of the risky cash flow with the
market portfolio.
In section I of this paper, we present the
results of a single-period analysis of a bank that
has both uninsured and insured deposits and
subordinated debt, as derived in Osterberg and
Thomson (1991). Section II extends our 1991
analysis to include the intended impact of
depositor-preference laws. In section III, we
■ 5 Title III of the Omnibus Budget Reconciliation Act of 1993
instituted depositor preference for all insured depository institutions by
amending Section 11(d)(11) of the Federal Deposit Insurance Corporation
Act [12 U.S.C. 1821(d)(11)]. At the time when national depositor preference was enacted, 29 states had similar laws covering state-chartered
banks, and 18 had depositor-preference statutes covering state-chartered
thrift institutions.
■ 6 For empirical studies of the impact of depositor-preference laws,
see Hirschhorn and Zervos (1990), Osterberg (1996), and Osterberg and
Thomson (1998).
■ 7 For simplicity, we assume that the deposit-insurance premium is
an end-of-period claim on the bank. This is equivalent to assuming that the
premium is subordinate to Bi and that, in effect, the bank receives coverage
while not necessarily paying the full premium. However, although this
assumption affects how the deposit-insurance subsidy enters into the
expressions in this paper and the actual size of the subsidy, it does not
qualitatively affect the results.
investigate the laws’ effects on the value of debt
capital and deposit guarantees when general
creditors behave strategically. We present conclusions and policy implications in section IV.
I. Banks’ Cost of
Capital and the
Value of Deposit
Insurance:
No Depositor
Preference
The following assumptions are used throughout
this paper: 1) the risk-free rate of interest is
constant; 2) capital markets are perfectly competitive; 3) expectations are homogeneous
respecting the probability distributions of the
yields on risky assets; 4) investors are risk averse
and seek to maximize the utility of terminal
wealth; 5) there are no taxes or bankruptcy
costs; 6) all debt instruments are discount
instruments, so the total promised payment
to depositors and subordinated debtholders
includes both principal and interest; and 7) the
deposit-insurance premium is paid at the end
of the period. 7
In this section, we present results from
Osterberg and Thomson (1990) for a bank with
insured deposits and uninsured deposits,
extended to include general creditors. The
FDIC charges a fixed premium of r on each
dollar of insured deposits. The total liability
claims against the bank, D, is the sum of the
end-of-period promised payments to the uninsured depositors, Bu , insured depositors, Bi ,
general creditors, G, and the FDIC, z ( r Bi ). We
assume that the FDIC underprices its deposit
guarantees on average, and that in the absence
of regulatory taxes (Buser, Chen, and Kane
[1981]), the FDIC provides a subsidy that
reduces banks’ cost of capital and increases
banks’ value.
Under these assumptions, the end-of-period
cash flows to insured depositors, Ybi , clearly
equal the promised payments to insured depositors, Bi , in every state. Therefore, whatever a
bank’s capital structure may be, the value,
expected return, and cost of one dollar of
insured deposits are defined as Vbi = R –1Bi ,
E (Rbi ) = r, and r + r , respectively.
12
of and the required rate of return on uninsured
deposits are
BOX 1
Definition of Notation
Bi = Total promised payment to insured depositors.
Bu = Total promised payment to uninsured depositors.
G = Total promised payment to general creditors.
r = Deposit-insurance premium per dollar of insured
deposits.8
z = Total promised payment to the FDIC ( r Bi ).
B = Total promised payment to depositors and the FDIC
(Bi + Bu +z).
S = Total promised payment to subordinated debtholders.
D = Total promised payment (Bi + Bu + G + S + z).
Ybi , Ybu, YG , Ys , Ye , and YFDIC = End-of-period cash flows to
insured depositors, uninsured depositors, general creditors,
subordinated debtholders, stockholders, and the FDIC.
Vbi , Vbu , VG , Vs , Ve , and VFDIC = Values of insured deposits,
uninsured deposits, general-creditor claims, subordinated
debt, bank equity, and the FDIC’s claim.
Vf = Value of the bank.
E (Rbi ), E (Rbu ), E (RG ), E (Rs ), and E (Re ) = Expected rates of
return on insured and uninsured deposits, general-creditor
claims, subordinated debt, and equity.
r = Risk-free rate (R =1 +r).
X = End-of-period gross return on bank assets.
F (X ) = Cumulative probability-distribution function for X .
l = Market risk premium.
COV (X, Rm ) = Systematic or nondiversifiable risk.
Rm = Return on the market portfolio.
CEQ(X ) = Certainty equivalence of X [E (X ) – lCOV (X, Rm )].
Vbu =R –1 {Bu [12F (D2S)]
(1)
+ [Bu /(D 2S )]CEQ D –S
0 (X )}
and
(2)
E (Rbu )=
12F (D2S )+[1/(D2S )]E D 0–S (X )
Vbu
–1.0.
Equation (2) shows that the cost of debt
(uninsured-deposit) capital is a function of
the bank’s systematic risk, as measured by
lCOV (X, Rm ); total promised payments to
depositors and the FDIC, (D2S); the probability
that losses will exceed the level of subordinated debt, F (D2S); and the risk-free rate of
return. Osterberg and Thomson (1990, 1991)
show that when the FDIC misprices its guarantees, the cost of uninsured deposit capital also
depends on the deposit mix, because underpriced (overpriced) deposit guarantees lower
(raise) the effective bankruptcy threshold for
senior claims, F (D2S), as well as the bankruptcy threshold, F (D). Furthermore, underpriced (overpriced) deposit guarantees increase
(decrease) the claims of uninsured depositors
relative to senior claims, Bu /(D2S), and relative to total claims, Bu /D. The size of this effect
is a function of the FDIC’s pricing error per
dollar of insured deposits and of the weight of
insured deposits in the senior creditor pool.
General Creditors
Uninsured Depositors
General creditors have the same priority of
claim as uninsured depositors; consequently,
they will have similar end-of-period cash flows.
End-of-period cash flows to uninsured depositors depend on the promised payment to the
uninsured depositors and on total promised
payments minus subordinated debt:
Ybu =
Bu
if X >D2S =Bi +Bu+G +z,
BuX/(D2S) if D2S >X > 0, and
0
if 0>X.
While total promised payments to debtholders
and the FDIC equal D, the effective bankruptcy
threshold for uninsured depositors is D less the
claims of subordinated debtholders. The value
YG =
G
if X >D2S =Bi +Bu+G +z,
GX/(D2S) if D2S >X > 0, and
0
if 0>X.
As before, total promised payments equal D,
and the effective bankruptcy threshold is
D2S. The value of and the required rate of
return on senior nondeposit debt are
(3) VG =R –1 {G [12F (D2S)]
+[G/(D2S)]CEQ D –S
0 (X )}
and
(4)
■ 8 For simplicity, we express the premium as a function of insured
deposits. The results of interest are not materially affected by adopting the
more realistic assumption that premiums are levied on total domestic
deposits, insured and uninsured.
12F (D2S )+[1/(D2S )]E D 0–S (X )
E (RG )=
21.0.
VG
13
Equation (4) shows that the cost of nondeposit debt (general-credit) capital is a
function of the same factors as uninsured
deposits, including the bank’s systematic risk,
lCOV (X, Rm ), total promised payments to
senior creditors and the FDIC, (D2S ), the
probability that losses will exceed the level of
subordinated debt, F (D2S), and the risk-free
rate of return. It also depends on the size of the
deposit-insurance subsidy.
Equityholders
The end-of-period cash flows accruing to
stockholders are
Ye =
X2D
if X >D, and
0
if D >X.
The value of equity and the expected return to
stockholders are
(7)
Subordinated Debtholders
Ve = R –1 {CEQD (X )2D [12F (D )] }
and
The end-of-period expected cash flows accruing to subordinated debtholders are
Ys =
S
if X >D,
X+S2D
if D >X > D 2S, and
0
if D 2S >X.
The value of the subordinated debt and the
required rate of return on subordinated debt
capital are
(5)
VS =R –1 {S [12F (D2S )]2D [F (D)2F (D2S )]
+CEQDD–S (X)}
(8)
+
S [12F (D2S )]2D[F (D)2F (D2S )]
EDD– S (X )
VS
VS
21.0.
Equations (5) and (6) show that the cost and
value of subordinated debt capital depend on
the probability of bankruptcy, F (D), the face
value of the subordinated debt, S, total promised payments, D, and the probability that
senior claimants will not be repaid in full,
F (D2S ). Note that the last two terms in equation (6) represent the claims of subordinated
debtholders in states where they are the residual
claimants.
Ve
21.0.
The net value of deposit insurance is the value
of the FDIC’s claim on the bank, that is, the
value of the FDIC’s premium less the value
of its deposit guarantee. In the absence of
depositor-preference laws, the end-of-period
cash flows to the FDIC and the value of its
position are
YFDIC =
E (Rs )=
ED (X )2D [12F (D )]
The FDIC’s Claim
and
(6)
E (Re )=
(9)
Z
if X >D, 2S,
(Bi+z)X/(D2S )2Bi
if D2S >X >0, and
2Bi
if 0>X , so that
VFDIC =R –1{z [12F (D 2S )]
B +z
+ i CEQ D 0–S (X )2Bi F (D2S )}.
D2S
Equation (9) shows that the net value of
deposit insurance is a function of the composition of the senior claims, the bank’s systematic
risk, the presence of junior debt claims in the
bank’s capital structure, the risk-free rate of
return, the effective probability of bankruptcy,
F (D2S ), the level of promised payments to
insured depositors, and the deposit-insurance
premium. In fact, equation (9) can be interpreted as showing that the equity-like buffer
provided by subordinated debt affects the value
of the FDIC’s position by changing the probability that put options corresponding to the
FDIC guarantee will be “in the money” at the
end of the period. Equation (9) also demonstrates that if deposit insurance is to be priced
fairly, VFDIC = 0, the premium will be influenced
14
by the degree to which the bank funds itself
with claims junior to insured deposits.
Osterberg and Thomson (1990) show that
the value of the uninsured bank is R –1CEQ 0(X ).
The value of the insured bank, Vf , equals the
uninsured bank’s value minus equation (9),
which is the value of the FDIC’s claim.
(10) Vf =R –1{CEQ 0(X )+Bi F (D 2S )2[(Bi +z)
/(D2S )]CEQ D –S
0 (X ) –z [12F (D2S )]}.
Equation (10) shows that the structure of a
bank’s debt (in terms of payment priority)
affects the value of the bank only through the
net value of deposit insurance to the bank.
To see this, note that Bi F (D2S ) 2 [(Bi +z)/
(D2S )]CEQ D 0–S (X ) is the value of FDIC guarantees, and z [12F (D 2S )] is the value of the
FDIC premium. If deposit insurance is correctly
priced (that is, the value of its guarantee equals
the value of its premium), then the structure of
a bank’s liability claims does not affect the
bank’s value.
Uninsured
Depositors
The end-of-period cash flows to uninsured
depositors depend on the promised payment
to uninsured depositors and on the total level
of promised payments minus subordinated debt
and the now-subordinated claims of general
creditors:
Ybu =
In this section, we rederive the results to
incorporate depositor preference, which subordinates the claims of general creditors to
those of uninsured depositors and of the FDIC.
As in section I, we assume that the FDIC charges
a flat-rate insurance premium of r on each dollar
of insured deposits, and that on average the
FDIC underprices its deposit guarantees.9 To
simplify the analysis, we assume that depositor
preference does not change total liability claims
against the bank, D.10 Under this assumption,
depositor-preference laws have no impact on
claims that are junior to deposits and generalcreditor claims.
■ 9 The results are qualitatively the same if the FDIC charges a
variable-rate premium, so long as the deposit guarantees are mispriced.
■ 10 The results for uninsured depositors, FDIC, and general-creditor
claims are qualitatively the same if we assume that depositor-preference
laws change the level of total promised payments (see Osterberg and
Thomson [1991, 1994]).
if X >B =Bi +Bu +z,
Bu X/B
if B >X > 0, and
0
if 0 >X.
While total promised payments to debtholders and the FDIC equal K, the effective
bankruptcy threshold for uninsured depositors
is B (=D 2G 2S ). The value of— and the
required rate of return on —uninsured
deposits are
(11)
Vbu =R –1 {Bu[1–F (B)] + (Bu /B)CEQ B0 (X )},
and
(12)
II. Banks’ Cost of
Capital and the Value
of the Insurance
Fund: Depositor
Preference
Bu
E (Rbu )=
12F (B)+(1/B)E B0 (X )
21.0.
Vbu
From the standpoint of uninsured deposit
capital, depositor-preference laws have the
same impact as a requirement that banks issue
subordinated debt. That is, when uninsured
depositors and the FDIC have claims in bankruptcy that are senior to those of general creditors, the effective bankruptcy threshold for
uninsured depositors is lowered from D2S
to D2G2S. For uninsured depositors (and, as
we shall see, for the FDIC), the pecking order
of more junior claims is irrelevant to the value
of their own.
To assess depositor preference’s impact on
the value of uninsured deposits, we control for
possible changes in total promised payments
by normalizing their expected cash flows by
the level of uninsured deposits, and compare
uninsured deposits in banks with and without
depositor-preference laws. We then separate
the expected cash flow to an uninsured deposit
(with a par value of one dollar) in the presence
of depositor-preference debt into two instruments. One is identical to the uninsured deposit
in section I; the other has the following endof-period payoffs and value:
15
DYbu = 0
if X >D 2S,
1–X/(D 2S)
if D 2S >X >B,
X/B 2X/(D2S)
if B >X >0, and
0
if 0 >X, so that
(D2S2B )
(13) DYbu = R –1[F (D2S )2F (B ) + B (D2S ) CEQ B0 (x)
2
1
CEQ DB –S (x) ] > 0.
D 2S
Equation (13) is positive; note that the first
term in the brackets is strictly greater than the
third term. Moreover, since by definition D2S >B,
the middle term is also positive. Therefore,
depositor preference must increase the value
of a dollar of uninsured deposits.
General Creditors
Under depositor preference, general-creditor
claims are junior to those of depositors and
the FDIC but senior to those of subordinated
creditors; hence, end-of-period cash flows to
general creditors are
YG = G
Equations (14) and (15) show that nondeposit debt (general credit) behaves like subordinated debt (equations [5] and [6]), except
that subordinated debt protects general creditors from loss. The value of general-creditor
claims depends on the effective bankruptcy
threshold, F (D2S ), the face value of their
claims, G, total promised payments to senior
claimants, B, and the probability that senior
claimants will not be repaid in full, F (B ). Note
that when earnings fall between B and D2S ,
general creditors are the residual claimants,
and theirs will behave like an equity claim.
Following the procedure used in the previous section, we construct the replicating portfolio for a general-creditor claim (with a par
value of one dollar) under depositor preference. With depositor preference, the expected
cash flow to such a claim is divided into one
part that is identical to the general-creditor
claim in section I, and a second that has the
following end-of-period payoffs and value:
DYG = 0
if X >D 2S,
(X2B)/G2X/(D2S ) if D2S >X >B,
2X/ (D2S )
if B >X >0, and
0
if 0 >X , so that
if X >D2S = Bi + Bu + G + z,
X2B
if D2S >X >B, and
0
if B >X.
The total promised payments to debtholders
and to the FDIC equal D, and the effective
bankruptcy threshold is D 2S. The value of
and the required rate of return on generalcreditor claims are
(16) DVG =R –1[
2
CEQ BD –S(X)2B [F (D2S )2F (B)]
G
CEQ D0 –S(X)
]< 0.
D2S
Equation (16) is unambiguously negative.
That is, depositor preference decreases the
value of a general-creditor claim.
(14) VG =R –1 {G [12F (D2S )]2B [F (D2S )2F (B)]
+CEQ DB –S (X )},
and
(15) E (RG )=
+
G [12F (D2S)2B [F (D2S)2F (B)]
VG
E DB –S (X)
21.0.
VG
The FDIC’s Claim
As before, the net value of deposit insurance
is simply the value of the FDIC’s claim on the
bank. Under depositor preference, the end-ofperiod cash flows to the FDIC and the value of
its position are
YFDIC = z
if
X >B,
(Bi+z)X/B–Bi
if
B >X > 0, and
–Bi
if
0 >X, so that
16
(17) VFDIC = R –1{z [12F (B )] +
Bi + z
CEQ B0 (X )
B
2Bi F (B)}.
As with uninsured deposits, the impact of a
depositor-preference law is indistinguishable
from a subordinated-debt requirement. Depositor preference affects the net value of the FDIC’s
claim by changing the senior claimants’ probability of loss and by altering the weight of the
FDIC in the pool of senior claims.
The change in the value of the FDIC guarantee on a one-dollar-par-value deposit is the
value of a security with the following cash flows
(where r = z/Bi ):
DYFDIC =0
r2(1+ r)X/(D2S )+1
if X >D 2S,
if D2S >X >B,
(1+r)X /B2(1+r)X/(D2S) if B >X >0, and
0
if 0 >X, so that
1+r
1
(18) VFDIC = R [F (D2S )2F (B )2 D2S CEQ DB–S (X)
D2S2B
+ B(D2S) CEQ B0 (X)] > 0.
Equation (18) is positive; to see this, note that
the first term in the brackets is strictly greater
than the second term. Since we assume that on
average the FDIC underprices its guarantees, its
claim on the bank is negative; hence, the size of
the FDIC subsidy is smaller under depositor
preference.
Finally, depositor preference affects the value
of the bank entirely through its effect on the net
value of deposit insurance.
(19) Vf = R –1 {CEQ 0 (X )2z [1–F (B)]+
Bi+z
CEQ B0 (X )
B
2Bi F (B)}.
Thus, if deposit insurance is always correctly
priced (that is, if its net value to the bank is
zero), depositor preference has no impact on
bank value. But it does change the fair value of
deposit insurance and so must be accounted for
when setting the premium.
■ 11 The decision to close a bank is based on one of two measures
of solvency: the incapacity to pay obligations as they mature or bookvalue, balance-sheet insolvency. Inability to renew nondeposit credits
could trigger insolvency under the maturing-obligations test
(see Thomson [1992]).
III. Banks’ Cost of
Debt Capital and the
Value of Deposit
Guarantees:
Depositor Preference
when General
Creditors Behave
Strategically
The results in section II assume that general
creditors do not respond to the subordination
of their claims under depositor preference.
However, in practice, general creditors of
insured depositories will respond to changes
in the priority of their claims and the higher
risk that results. At the very least, general
creditors will charge the depository institution
a higher rate of interest to compensate for the
increased risk of loss. As nondeposit funds
become more expensive relative to deposits,
institutions will lessen their funding in nondeposit markets, thus reducing the loss buffer
afforded to uninsured depositors and the FDIC
by nondeposit creditors.
Senior nondeposit creditors might also
respond to depositor preference by reducing
the average maturity of their claims. This
response increases creditors’ ability to “run” on
the depository institution if its condition deteriorates. In fact, financially distressed institutions
may find it difficult or impossible to issue unsecured nondeposit claims. This response by
nondeposit creditors to depositor preference
has two implications. First, if nondeposit
creditors can effectively exit a troubled institution before it is closed, little or no loss cushion will be afforded to uninsured depositors
and the FDIC. Second, the failure of nondeposit
creditors to renew their claims could trigger
a liquidity crisis that causes the institution to
be closed.11
The third option for unsecured general
creditors is to take collateral against their claim.
By becoming secured creditors, they will have
transformed their claim into one that is senior
(to the extent of the collateral) to deposit claims.
This, in turn, will have two effects on the
claims of uninsured depositors and the FDIC.
First, the loss buffer afforded by generalcreditor claims is reduced. Second (and more
importantly), the general asset pool available
to pay unsecured claims is also reduced. If
enough general-creditor claims take collateral,
the total loss exposure of the FDIC and uninsured depositors could increase.
17
Structural
Arbitrage
(20)
The static nature of our model does not allow
us to study the dynamic reaction of general
creditors to depositor-preference laws directly.
However, we can examine the implications of
structural arbitrage by general creditors through
its impact on the cash flows accruing to each
class of claimant. Under the assumption that
general creditors effectively collateralize their
claims on the bank, we can show the unintended effect of depositor-preference laws
on the cost of capital for banks and on the
FDIC’s claim.
As in section I, we assume that the FDIC
charges a flat-rate insurance premium of r on
each dollar of insured deposits and that, on
average, the FDIC underprices its deposit guarantees. The total liability claims against the
bank, D, are the sum of the end-of-period
promised payments to uninsured depositors,
Bu , insured depositors, Bi , general creditors,
G, subordinated debtholders, S, and the FDIC,
z (= r Bi ). As in the previous section, we
assume that total claims, D, are not affected by
depositor preference and general creditors’
responses to it.
Uninsured
Depositors
(
)
{CEQ GD –S (X )2G [F (D2S )2F (G )]} ,
and
(21)
Ebu = 12F (D2S )
Vbu
1
D –S
D–S {E G (X )2G [F (D2S )2F (G )]} 21.0.
+
Vbu
From the standpoint of uninsured deposit
capital, general creditors’ strategic behavior has
rendered depositor claims junior to their own.
To isolate the de facto impact of depositor
preference on the value of uninsured deposits
in this case, we control for possible changes in
total promised payments by normalizing
expected cash flows at the level of uninsured
deposits, and compare uninsured deposits in
banks in the presence and absence of depositorpreference laws. We then separate the expected
cash flow to an uninsured deposit (with a par
value of one dollar) in the presence of depositor-preference debt into two instruments: one
that is identical to the uninsured deposit in
section I, and a second that has the following
end-of-period payoffs and value:
DYbu = 0
The end-of-period cash flows to uninsured
depositors depend on the promised payment
to uninsured depositors, the total level of
promised payments minus subordinated debt
and claims, and the claims of general creditors:
Ybu = Bu
B
Vbu =R –1 Bu [12F (D2S )+D2uS
if X >D 2S,
Bu(X2G)/(D 2S)
if D2S >X >G, and
0
if 0 >X.
While the total promised payments to
debtholders and the FDIC equal D, the effective
bankruptcy threshold for uninsured depositors
is (D 2S ). The value of and the required rate of
return on uninsured deposits are
(22)
if X >D 2S,
2G/(D 2S)
if D2S >X >G, and
2X/(D 2S)
if G >X >0, so that
DVbu=[R(D2S)] –1{2G [F (D2S )2CEQ G0 (X)]} < 0.
Equation (22) is unambiguously negative.
Hence, a potential unintended effect of
depositor-preference laws is to reduce the
value of uninsured depositor claims on
the bank.
18
General Creditors
(25)
The intended effect of depositor preference is
to make general-creditor claims junior to those
of depositors and the FDIC, but senior to those
of subordinated creditors. However, the de facto
effect of depositor preference may be to make
general-creditor claims senior to all others.
Under this scenario, the end-of-period cash
flows to general creditors are
DYG = G
CEQ D0 –S (X )
DVG =R 21 {[F (D2S )2F (G )]2[
D2S
–
CEQ G0 (X )
] } > 0.
G
Whether the value of general-creditor claims
increases or decreases depends, in this case,
on whether the difference between the first two
bracketed terms in (25) is larger than the difference between the second two bracketed terms.
if X >G,
X
if G >X >0, and
0
if 0 >X .
The total promised payments to debtholders
and to the FDIC equal K, and the effective
bankruptcy threshold for general creditors is
D2B2S =G. The value of and the required rate
of return on general creditor claims are
The FDIC’s Claim
As before, the net value of deposit insurance
is the value of the FDIC’s claim on the bank.
Under depositor preference, end-of-period
cash flows to the FDIC and the value of its
position are
YFDIC = z
(23) VG =R –1{G [12F (G )]+CEQ G0 (X )},
and
(24)
E (RG )=
G [12F (G )]+E G0 (X )
21.0.
VG
Equations (23) and (24) show that the value
and return on general-creditor claims depend
only on the level and variability of cash flows
and the size of G. The presence and structure
of other claims on the bank do not affect the
valuation of such claims because we have
assumed that general creditors have de facto
secured the most senior claim on the bank.
Following the procedure used in the previous
sections, we construct the replicating portfolio
for a general-creditor claim (with a par value
of one dollar) under depositor preference.
The expected cash flow to a such a claim is
divided into one part that is identical to the
general-creditors claim in section I, and another
that has the following end-of-period payoffs
and value:
DYG = 0
(26)
if X >D 2S,
(Bi + z 2G)
X/(D 2S) 2Bi
if D2S >X >G, and
2Bi
if G >X, so that
VFDIC =R –1 {z [12F (D2S )]+
B i +z 2G
D2S
CEQ GD –S (X )2Bi F (D2S )}.
As with uninsured deposits, depositorpreference law’s impact on the FDIC’s claim
on the bank depends on the degree to which
general creditors engage in structural arbitrage.
The change in the value of the FDIC’s guarantee on a one-dollar-par-value deposit is the
value of a security that has the following cash
flows (where r = z/Bi ):
DYFDIC = 0
GX/(Bi (D2S)
if X >D2S ,
if D2S >X >G,
2(1 +r)X/(D2S ) if G >X >0, and
0
if 0 >X, so that
if X >D2S,
12X/(D2S )
if D2S >X >G, and
X/G2X/(D2S )
if G >X > 0, so that
(27)
G
VFDIC =[R (D2S )]21[2 B CEQ GD –S (X )
i
2(1+ r)CEQ G0 (X )]<0.
19
Equation (27) is clearly negative. Hence, a
possible unintended outcome of the national
depositor-preference law is to reduce the value
of the FDIC’s claim on the bank—that is, to
increase the value of the FDIC’s guarantees.
Finally, depositor preference influences the
value of the bank entirely through its effect on
the net value of deposit insurance:
(28)
the national depositor-preference law may
actually decrease the value of depositor and
FDIC claims—that is, it may increase the value
of deposit guarantees. Ultimately, whether this
unintended effect of depositor-preference law
will dominate is an empirical issue.
References
Vf =R –1[CEQ 0 (X )2z [12F (D2S )]2
Bi +z2G
CEQ GD –S (X ) +Bi F (D2S )].
2 D2S
As in the previous case, if deposit insurance is
always priced correctly (that is, if its net value to
the bank is zero), it has no impact on bank value.
However, depositor preference does change the
fair value of deposit insurance and so must be
accounted for when setting the premium.
IV. Conclusions
Using the cash-flow version of the capital-asset
pricing model, we show how depositorpreference laws affect the value and pricing of
claims on insured banks. The intended effect of
depositor preference is to change the bank’s
capital structure in a way that increases the value
of uninsured deposit claims and reduces the size
of the FDIC subsidy. Under the assumptions in
this paper, all general creditors would see the
value of their one-dollar-par claims reduced, to
the benefit of the FDIC and uninsured depositors.
Under less restrictive assumptions, other claimants
junior to depositors would also see the value of
their claims reduced. Overall, the intended effect
of a depositor-preference statute would be the
same as that of a mandatory subordinated debt
requirement.
Depositor-preference laws, however, have
another possible effect. Unlike subordinated-debt
holders, general creditors can act to offset the
statutory junior status of their claims.12 In its most
extreme form, structural arbitrage by general
creditors can de facto render depositor and FDIC
claims junior to those of general creditors. Hence,
■ 12 For example, holders of general-creditor claims could conceivably restructure their claims by taking collateral, thereby improving their
position relative to depositors and the FDIC. Hirschhorn and Zervos (1990)
find that for thrifts in states with depositor-preference laws, general creditors
are more likely to be collateralized; hence, in those states these laws give
depositors little protection.
Buser, Stephen A., Andrew H. Chen, and
Edward J. Kane. “Federal Deposit Insurance, Regulatory Policy, and Optimal Bank
Capital,” Journal of Finance, vol. 36, no. 1
(March 1981), pp. 51–60.
Carnell, Richard Scott. “A Partial Antidote to
Perverse Incentives: The FDIC Improvement
Act of 1991,” in Morin Center for Banking
Studies, Boston University School of Law,
Annual Review of Banking Law, vol. 12.
Boston, Mass.: Butterworth Legal Publishers,
1993, pp. 317–71.
Chen, Andrew H. “Recent Developments in
the Cost of Debt Capital,” Journal of
Finance, vol. 33, no. 3 (June 1978),
pp. 863 –77.
DeGennaro, Ramon P., and James B.
Thomson. “Capital Forbearance and Thrifts:
Examining the Costs of Regulatory Gambling,” Journal of Financial Services
Research, vol. 10, no. 3 (September 1996),
pp. 199 –211.
Hirschhorn, Eric, and David Zervos. “Policies to Change the Priority of Claimants: The
Case of Depositor Preference Laws,” Journal
of Financial Services Research, vol. 4, no. 2
(July 1990), pp. 111–26.
Kane, Edward J. The Gathering Crisis in
Federal Deposit Insurance. Cambridge,
Mass.: MIT Press, 1985.
________. The S&L Insurance Mess: How Did It
Happen? Lanham, Md.: The Urban Institute
Press, 1989.
20
Osterberg, William P. “The Impact of
Depositor Preference Laws,” Federal Reserve
Bank of Cleveland, Economic Review, vol. 32,
no. 3 (1996 Quarter 3), pp. 2 –11.
________, and James B. Thomson. “Deposit
Insurance and the Cost of Capital,” in
Andrew H. Chen, ed., Research in Finance,
vol. 8. Greenwich, Conn.: JAI Press, 1990,
pp. 255–70.
________, and ________. “The Effect of Subordinated Debt and Surety Bonds on the Cost
of Capital for Banks and the Value of Federal
Deposit Insurance,” Journal of Banking and
Finance, vol. 15, nos. 4–5 (September
1991), pp. 939 –53.
________, and ________. “Depositor Preference
and the Cost of Capital for Insured Depository Institutions,” Federal Reserve Bank
of Cleveland, Working Paper no. 9404,
April 1994.
________, and ________. “Depositor Preference
Legislation and Failed Bank Resolution
Costs,” in Federal Reserve Bank of Chicago,
Payments Systems in the Global Economy:
Risks and Opportunities, proceedings of the
34th Annual Conference on Bank Structure
and Competition, May 1998, pp. 226–46.
________. “A Market-Based Approach to
Reforming Bank Regulation and Federal
Deposit Insurance,” in George G. Kaufman,
ed., Research in Financial Services: Private
and Public Policy, vol. 4. Greenwich, Conn.:
JAI Press, 1992, pp. 93 –110.
Todd, Walker F. “New Discount Window
Policy Is Important Element of the FDICIA,”
Banking Policy Report, March 1, 1993, pp. 1
and 11–22.
21
Household Production
and Development
by Stephen L. Parente, Richard Rogerson, and Randall Wright
Stephen L. Parente is an assistant professor of
economics at the University of Illinois; Richard
Rogerson and Randall Wright are professors of
economics at the University of Pennsylvania.
The authors thank Jess Benhabib, Jeremy
Greenwood, Howard Pack, Edward Prescott,
Doug Collins, Cheryl Doss, and Norman Loayza,
as well as participants in seminars and conferences at the NBER Summer Institute; the SED
meetings in Oxford; the Federal Reserve Banks
of Philadelphia and Kansas City; the University
of Victoria; the University of California–
Riverside; Pennsylvania State University;
the University of Southampton; New York
University; the University of British Columbia;
MIT; and Harvard University, for their comments.
Introduction
Differences in standards of living across countries are large. For example, Summers and Heston (1991) indicate that income per worker is
around 30 times higher in the richest countries
than it is in the poorest countries.
Why are differences in living standards so
big? One position is that some countries have
relatively low income levels due to their relatively low stocks of capital; this is particularly
true if we interpret capital generally to include
human and other intangible capital (see Mankiw,
Romer, and Weil [1992]). Of course, this raises
the question, why do some countries have such
low capital stocks in the first place? One suggestion is that these countries are burdened with
policies that distort agents’ incentives to accumulate capital, policies that will be referred to
here as barriers to capital accumulation.
This paper analyzes the effects of such
barriers quantitatively. Compared to previous
studies that have analyzed the effects of such
■ 1 For example, Parente and Prescott (1994) and Chari, Kehoe, and
McGrattan (1996).
policies on relative levels of income in the
neoclassical growth model,1 the key difference
in this study is that we explicitly incorporate
nonmarket activity—that is, household production—into the analysis. We argue that distinguishing between economic activity in market
and nonmarket sectors may go a long way
toward understanding international differences
in capital and income.
The essence of our argument is as follows:
First, the nature of the development experiences
that we describe leads us to explain differences
in per-worker income levels across countries
(rather than differences in growth rates). The
question, therefore, is how much of the
observed differences in income levels can be
attributed to empirically realistic barriers to
capital accumulation? It is well known that the
standard neoclassical growth model accounts
for very few of these differences. However,
the effects of such policies can be significantly
larger when home production is included in the
model, at least for certain parameter values (in
particular, values that imply that capital is less
important in nonmarket than in market production, such that nonmarket- and market-produced
goods are relatively close substitutes). For
example, in a standard model without home
22
production, the distortionary policy must be
about 100 times larger in one country for it to
have one-tenth the income of another country,
while in one (admittedly extreme) version of
our home-production model, the distortionary
policy need be only about three times bigger.
In models with household production, agents
are generally more willing to shift resources out
of market activity in response to policy distortions. Intuitively, policies that affect capital
accumulation may also influence the mix of economic activity in market and nonmarket sectors,
and so policy distortions can have significant
effects. In the standard model without home
production, the policy distortion required to
generate a given income difference is so large
because the time agents spend working in the
market does not depend on the size of the distortion (given functional forms consistent with
balanced growth). Hence, in that model, crosscountry differences in output per worker are
entirely attributed to differences in capital per
worker. In the home-production model,
although these same policies may not affect total
hours worked, they generally do affect how
hours are allocated between the market and
nonmarket sectors. As individuals change their
allocation of time spent in market work and in
home work, differences in output per person
will be due to both differences in capital and in
market hours per worker.
As Parente and Prescott (1994) and Chari,
Kehoe, and McGrattan (1996) have noted, an
augmented neoclassical growth model without
household production—but with capital broadly
defined to include tangible and intangible
capital— can go quite far in accounting for
differences in income with reasonably sized
barriers, if one is willing to assume that total
capital’s share in the production function is
large. Such models, however, imply a large
amount of unmeasured capital and investment.
Household production is a complementary
extension of the neoclassical model, in that a
sizable fraction of the observed differences in
income across countries can be accounted for
in a model without intangible capital. If we
include both intangible capital and home production, the amount of intangible capital and,
hence, the role assigned to unmeasured investment, will be smaller.
The model with intangible capital and the
model with home production both entail
unmeasured output in the economy. However,
■ 2 See Greenwood and Hercowitz (1992), Behabib, Rogerson, and
Wright (1992), and McGrattan, Rogerson, and Wright (1997), for example,
for applications of home production in business-cycle theory.
in the home-production model, this unmeasured output takes the form of consumption
rather than investment. Furthermore, the model
without home production predicts the same
fraction of unmeasured output in rich and poor
countries, while the home-production model
predicts a greater fraction of unmeasured output
in poor countries. Hence, the home-production
model predicts that the true differences in output, and especially in consumption, are smaller
than those reported in the National Income and
Product Accounts because nonmarket production and consumption are relatively more
important in poor countries. This helps us to
understand how individuals can survive on the
amount of consumption reported in the official
data in the poorest countries—an issue raised,
for example, by Lucas (1988). It also allows us to
compute the welfare implications of policy differences and, thus, of output differences in a
way that explicitly recognizes nonmarket activity and its importance in poorer countries.
It may be worth mentioning at this point an
analogy between the approach to development
economics adopted here and modern businesscycle theory. In that literature, one attempts to
identify and measure impulses (the underlying
sources of fluctuations, such as technology
shocks, changes in monetary policy, and so on)
and then study the extent to which these
impulses are amplified or propagated by different economic models. For example, one might
ask, what fraction of observed business-cycle
fluctuations can be accounted for with a given
impulse and model? In this paper, we take as a
maintained (if presumably counterfactual)
hypothesis that countries differ only with
respect to their barriers to capital accumulation.
We establish reasonable magnitudes for these
barriers, and then we ask, what fraction of the
observed income differences can we account
for? The point of this analysis is that the answer
to the question changes once one recognizes
that much economic activity takes place outside
the formal market. Likewise, answers to several
questions in business-cycle theory change once
one incorporates home production into otherwise standard models.2
The rest of the paper is organized as follows:
Section I reviews some basic development facts.
Section II documents how the standard neoclassical growth model fails to account for these
facts, given empirically plausible parameter values. We show how this model, augmented to
include a second form of capital, can account
for these facts, but also can predict a large
amount of unmeasured capital and investment.
Section III introduces home production into the
23
F I G U R E
1
Relative Income, Richest Countries
to Poorest Countries
Ratio
40
35
30
25
20
15
10
5
0
1960
1965
1970
1975
1980
1985
SOURCE: Summers and Heston (1991).
basic neoclassical model (without the second
form of capital) and reports the quantitative
impact of size differences in the barrier for
several parameterizations of the model on
observable variables and on welfare. Section IV
integrates the model with two types of capital
and with household production. Section V considers
evidence supporting the view that home production is relatively important in less-developed
economies.3 Section VI contains some brief
concluding remarks.
I. Key Development
Facts
In this section we briefly present some key
development facts (more detailed discussions
■ 3 The idea that household production may be important to understanding economic development is not new. Kuznets (1960), for example,
noted that nonmarket activities are more important in relatively poor
nations. Eisner (1994) attributes the difference between the true and
reported outputs to home production. Previous attempts to model household production in the context of economic development include Hymer
and Resnick (1969) and Locay (1992), but these are not quantitative,
dynamic, general-equilibrium models. Easterly (1993) studies an endogenous growth model that can be interpreted as having a formal and informal
sector, although it could just as well be interpreted as a one-sector model
with two types of capital.
can be found in Parente and Prescott [1993,
1994]) which dictate our choices of questions
and modeling strategies. Some researchers have
concluded that a theory of relative income levels, as opposed to a theory of growth-rate differentials, is appropriate for understanding the
pattern of economic development. Since an
exogenous growth model of relative income
levels is the framework of this paper, we motivate our choice by describing the relevant data.
Let yt measure gross domestic product (GDP)
per worker in a country at date t divided by
GDP per worker in the United States at date t,
computed at world prices. For the 102 countries
in Summers and Heston (1991) with at least one
million in population for which the data is complete between 1960 – 88, figure 1 plots the ratio
of the average yt in the five richest countries to
the average yt in the five poorest countries. First,
notice that the GDP disparity is big—the richest
five countries are about 30 times richer than the
poorest five. Second, observe that this disparity
has not increased. The ratio remains essentially
the same over the period. (The standard deviation of the income distribution increases some,
from about 1.25 to 1.50, with some of the mass
spreading from the center to the tails.) In
addition, while the rich got richer, so did the
poor: with rare exceptions, all countries grew,
suggesting no absolute poverty trap. The average
annual growth rate in the sample is 1.9 percent.
Although it cannot be seen in figure 1, individual countries have moved within the distribution, suggesting no relative poverty trap: there
have been both miracles and disasters.
Let us take as a base y;, set equal to 10 percent of per capita GDP in the United States in
1985. Figure 2 plots the year in which each
country achieved this level against the number
of years it took that country to double its per
capita output (that is, to go from y;to 2 y;).
Countries that achieve an income level of y;
relatively early will take a longer period of time
to double their income, while countries that
achieve an income level of y; later can double
their income much more rapidly. (This does not
depend crucially on the choice of the base, and
a similar pattern emerges for other values of y;.)
Hence, while the frontier is growing at a given
rate, if a country lags significantly behind, it is
possible to make rapid advances toward the
frontier. This suggests that some countries have
a policy or a set of institutions, perhaps, that
keeps their income relatively low, but they are
capable of catching up somewhat if the policy is
eliminated or ameliorated.
24
F I G U R E
2
Years to Grow from $2,000 to $4,000 Per Capita Income
versus Year Reached $2,000
Years to grow from $2,000 to $4,000
80
70
■
Ire ■
NZ
Belg ■
60
UK
■
Neth ■
50
Austal ■
40
US
■
■
Hung ■
USSR
■ Aus
Spn ■ ■■ Ger
Ita ■ ■ Fra
Den ■
Swit ■
Chil ■
Swed ■
Arg ■
■■ ■
Cze
Can ■
30
S.Afr ■
Jpn
Grc
Fin ■
■
Nrw
20
Jam
10
0
1810
1830
1850
1870
1890
1910
Year reached $2,000
Bzl
Col ■
■
1930
Peru ■
Tur
■ Mal
Pol ■■ Irq
■
Mex ■
CR
Ecd
Port ■ Irn ■ ■ ■■■
Rom
■■
■
Jor
Bulg ■ ■■■■
Nam
■
■■ ■ Twn
HK
PR
Yug
Syr
Pan
1950
1970
1990
a
a. Measured in 1990 U.S. dollars.
SOURCE: Parente and Prescott (forthcoming 2000).
These facts influence our choice of questions
and models. In an endogenous growth model,
differences in policies translate into differences
in growth rates, but the data indicate that
growth-rate differences are not permanent, as
income levels across countries do not diverge
over the postwar period. Even if we allow
policies and (therefore) growth rates to change
over time, an endogenous growth model cannot
clearly explain the fact that countries that
achieve a given base income later are able to
double their income more quickly. Therefore, it
is reasonable to adopt an exogenous growth
model—that is, to assume that countries grow
at the same average rate—and to ask what produces the observed differences in relative
income levels.4 In this model, countries that are
behind the frontier because of a particular policy
can indeed move up rapidly within the income
distribution once the policy is removed. It is also
■ 4 Exogenous growth, incidentally, does not mean that a country can
realize increases in output without undertaking any action. Parente and
Prescott (1994) show that the equilibrium behavior of a model in which
firms choose whether to adopt better technologies over time and in which
the stock of knowledge that firms can adopt increases exogenously, is
equivalent to the neoclassical growth model augmented with a second form
of capital.
desirable to consider policies for which we have
some quantitative information. This will lead us
to model barriers to capital accumulation in a
particular way, as we discuss below.
II. Background
The starting point of our analysis is the standard
one-sector growth model. Assume an infinitely
lived representative agent with preferences
given by
(1)
¥
^ b t [logct + a log(1–nt )] ,
t=0
where ct denotes consumption, nt denotes time
spent working at date t, and b Î (0,1) denotes
the discount factor. The representative agent is
endowed with one unit of time in each period
and k 0 units of capital at t =0. A constant-returnsto-scale production function uses capital and
labor to produce output
(2)
yt= A k tq [(l+ g ) t nt ]1–q,
25
F I G U R E
3
Relative Investment Shares,
Rich vs. Poor Countries
Percent
2.5
2.0
Purchasing-power-parity prices
1.5
1.0
Domestic prices
0.5
0
1965
1970
1975
1980
1985
1990
SOURCES: Summers and Heston (1991), International Monetary Fund (1994)
where exogenous, labor-augmenting technological change occurs at rate g per period.
Output in each period is divided between
consumption and investment,
(3)
ct + xt ² yt .
Capital accumulation is represented by
(4)
kt+1 = (1– d )kt +xt ,
where d Î (0,1) is the depreciation rate.
One approach to studying the implications of
the neoclassical growth model for development
is to view each country as a closed economy,
described by the same preferences and technology, and to look for policies that differ across
countries and that may affect relative levels of
output. Since the key economic decision in the
model is the consumption–savings decision, it is
natural to look for policies that distort incentives
for agents to accumulate capital. There are many
candidate factors, ranging from taxation and
regulation to fear of confiscation. Two policies
that have been studied in the literature are capital
income taxation and policies affecting the relative price of capital goods, which we refer to
as barriers to capital accumulation. From the
perspective of modeling relative income, these
two policies have similar effects. However, there
are reasons to believe that barriers that distort
the relative price of investment goods may be a
more promising route.
If cross-country income differences are
explained by differences in tax rates, then per
capita income and taxes should be negatively
correlated; however, the data do not show such
a relationship (see Easterly and Rebelo [1993]).
If one assumes that barriers to capital accumulation are the source of income differences, then
there should be a negative correlation between
per capita income and the price of investment
relative to consumption goods. Jones (1994)
documents differences in the price of equipment relative to consumption goods across
countries and does indeed find a strong negative correlation between this variable and per
capita output.
Since the two policies have different implications for prices, they have different implications
for the investment–income ratio. Empirically,
investment shares measured using domestic
prices display no correlation with per capita output, whereas investment shares computed using
purchasing-power-parity prices show a positive
correlation with output per capita. Figure 3
plots the ratio of investment share in rich countries to the investment share in poor countries,
computed with both domestic prices and
with purchasing-power-parity prices. We argue
that models with barriers that distort the relative
price of capital are better able to match this
observation.
We focus on policies that affect the price of
investment relative to consumption goods,
parameterized by changing capital accumulation
(equation [4] ) to
(5)
kt+1 = (1– d )kt + xt ,
p
where p is the size of the barrier. If p =1, then
one unit of consumption can be turned into one
unit of capital, while in a country with p > 1,
each unit of consumption invested yields only
(1/p < 1) units of capital. Thus, p is the relative
price of investment goods. Jones’ (1994)
26
T A B L E
1
Differences in Relative
*
Income, y*/ yUS
U =1/4
U =1/3
U =1/2
U =2/3
U =3/4
p=2
0.79
0.71
0.50
0.25
0.13
p=3
0.69
0.58
0.33
0.11
0.04
p=4
0.63
0.50
0.25
0.06
0.02
p = 10
0.46
0.32
0.10
0.01
0.001
SOURCE: Authors’ calculations.
evidence not only gives us reason to believe that
differences in barriers may be the source of
income differences, it also allows us to establish
reasonable estimates of the magnitude of p.
Jones reports a range of equipment relative to
consumption goods prices between 1 and 4
across countries, with the United States normalized to 1.5
Pursuing the business-cycle analogy, we now
have a more or less quantifiable impulse for the
phenomenon in which we are interested, and
we can now think about asking, how much of
the observed income disparity can be accounted
for by variations in p ? Although we do not
believe that policies distorting the prices of
■ 5 Restuccia and Urrutia (1996) find a range closer to 12. Like Jones
(1994), they compute the ratio of the price level of investment to the price
level of consumption goods, taken from the Penn World Tables; unlike Jones,
they consider all investment goods, rather than a subcategory, and use both
benchmarked and unbenchmarked countries. However, the big difference
seems to be due to revisions of the price data in the Penn World Tables
between PWT and PWT5.6. In another sense, the range of 4 may be too large.
According to PWT5.6, the prices of consumption goods vary much more
across countries than prices of investment goods (that is, it is not that
computers are expensive in poor countries, but that haircuts are cheap).
It appears that differences in the price of consumption goods account for
about half of the variation in these relative prices, suggesting a barrier
closer to 2.
■ 6 Note that, even though this economy has a distortionary policy, one
can characterize a competitive equilibrium by solving an augmented social
planner’s problem, where the planner faces the law of motion for kt that
includes the barrier kt = (1–d)k t + xt /p t .
investment goods are the only factor accounting
for cross-country income differences, it is still of
interest to examine the effects that can be generated as a function of p , given a particular model.
Given p , the unique equilibrium in the model
has the property that, starting from any k 0 >0,
we converge to a balanced growth path, along
which output, consumption, investment, capital,
and wages all grow at the same rate g , while the
rate of return to capital and hours worked are
constant.6 For the most part, we will focus on
balanced growth paths, and in particular on the
relative level of balanced-growth-path output
across economies with different values of p
(although we will also take into account transitions from one balanced growth path to another
when we analyze welfare). With our functional
forms, it is straightforward to characterize the
balanced growth path as a function of p . First,
the fraction of time devoted to work is independent of p . Second, given that p =1 in the United
States, the relative capital stock and output in
an economy with a barrier of size p will be
(6)
k*
y*
= p –1/(1–U) and
= p –U/(1–U) .
*
*
k US
y US
Does this yield a good theory of development? To answer this question, one must say
something about parameter values. Table 1
reports the relative output differences generated
by differences in barriers from p =2 to p =10 for
various values of U. If k is interpreted as physical
capital, we are led to consider a value of U
between 1/4 and 1/3. For these parameter values, the model accounts for very little of the
observed differences in per capita income; for
example, with U = 1/3, U.S. output is only twice
as high as output in an economy with p = 4,
and only three times as high as an economy with
p = 10. Recall that the data indicate that U.S.
output per worker is 30 times higher than output
per worker in the poorest countries. To generate
output differences of 30 with U =1/3, we would
need a barrier of p = 900. This model is off by
orders of magnitude.
Table 1 provides results for higher values of
U, which, as one can see, allow the model to
account for much larger differences in relative
income. For example, if U =2/3, we can generate output differences of 30 with a barrier of
about p = 5.5. To rationalize such higher values
of U, several researchers (including Mankiw,
Romer, and Weil [1992], Parente and Prescott
[1994], and Chari, Kehoe, and McGrattan [1996])
27
T A B L E
uct accounts corresponds to (ct +xkt ) in the
model. We assume the two capital stocks evolve
according to:
2
Model with
Unmeasured Capital
(8) kt+1 =(1– dk )kt +xkt /p
Uk
Uz
xz
c+xk
xk
c+xk
k
z
0.10
0.57
0.76
0.13
0.18
0.20
0.47
0.55
0.24
0.43
0.30
0.37
0.39
0.32
0.82
0.40
0.27
0.26
0.38
1.50
0.50
0.17
0.15
0.44
3.00
0.60
0.07
0.05
0.48
9.00
SOURCE: Authors’ calculations.
have discussed expanding the notion of capital
beyond physical capital to broader notions,
including human and organizational capital.
However, some issues must be faced if one goes
this route. First, the values of p that we infer
from empirical work are based on data for
physical capital, and it is unclear to what extent
barriers of this size apply to other types of
investment. For example, in the case of human
capital, a substantial component of accumulation takes place in the formal education sector,
which is heavily subsidized in many countries,
especially in poor countries. Second, since current national-income-accounting procedures do
not recognize capital other than physical capital, and investments in intangible capital (with
the exception of some education expenditures)
are not measured, assuming high values of U
implies that a large amount of capital and investment will go unmeasured.
To illustrate this point explicitly, we modify
the previous model to include an intangible
capital good, z. Let investment in physical and
in intangible capital be denoted by xk and xz ,
respectively, so that we have
t
1– Uk – U z .
z
(7) ct +xkt +xzt ²yt =AktUk zU
t [(1+ g) nt ]
The empirical counterpart of yt is no longer
output as reported in the National Income and
Product Accounts, due to the unmeasured
investment; rather, output in the national prod-
and
(9) zt+1 =(1– dz )zt +xzt /p .
It is a straightforward generalization of the
standard model to characterize the balanced
growth path for this model as a function of p .
Table 2 presents summary statistics for several combinations of Uk and Uz that sum to 2/3,
meaning that the p needed to generate the
observed international income differences is
5.5.7 The third column reports unmeasured
investment as a fraction of measured output; the
fourth column reports measured investment as a
fraction of measured output; and the last column reports the ratio of the two capital stocks. If
Uk =0.40, for example, unmeasured investment
is only 26 percent of measured output. However, such a value for Uk implies that measured
investment equals 38 percent of measured output—nearly twice as high as in the U.S. data. To
match the measured investment–output ratio, Uk
cannot be greater than 0.20; for such values of
Uk , the implied value of unmeasured investment exceeds half of measured output, and the
unmeasured capital stock is well over double
the measured capital stock.8
This discussion is not meant to suggest that
models with higher capital shares are necessarily
inconsistent with the data; after all, it is tautological to say that we do not have measures of
unmeasured investments.
We believe, however, it does suggest that
it may be worthwhile to consider other approaches.
■ 7 Other parameters are calibrated as follows: d k = dz = 0.06; y is
set to achieve 2 percent growth per year; b is set to achieve a real interest
rate of 4.5 percent; the preference parameter a is set so that the fraction of
time spent working is 1/3.
■ 8 Chari, Kehoe, and McGrattan (1996) set Uk =1/3, making their
model inconsistent with the ratio of measured investment to measured output. Mankiw, Romer, and Weil (1992) likewise set Uk =1/3, but their model
is not inconsistent with this observation, since they calibrate other parameters to match the measured investment–output ratio; however, this implies
a real rate of interest in excess of 10 percent. Parente and Prescott (1994)
calibrate to a real rate of return of 4.5 percent and a measured investmentoutput ratio of 20 percent, implying that Uk = 0.19 and that unmeasured
investment is 41 percent of measured output. This is lower than reported in
table 2 because Parente and Prescott’s calibration implies a lower depreciation rate on intangible capital of 3.5 percent. Their ratio of unmeasured to
measured capital, however, is still quite large.
28
The remainder of this paper outlines an alternative but complementary framework—the homeproduction model. This model helps to account
for cross-country income differences, but at the
same time has implications differing from the
previous models. For example, while both
approaches imply unmeasured output, the
model with intangible capital and without home
production emphasizes unmeasured investment, whereas the home-production model
emphasizes unmeasured consumption. Although
it is not apparent from table 2, an important
point for future reference is that the model with
intangible capital and without home production
implies that unmeasured investment as a fraction
of measured output is independent of p .9 Thus,
high- and low-distortion economies have the
same ratios of unmeasured investment to measured output, and so the difference in measured
output across countries accurately reflects the
difference in total (measured plus unmeasured)
output. This will not be true for the homeproduction model.
researchers in macroeconomics have found this
useful in accounting for high-frequency aspects of
the data (for example, Benhabib, Rogerson, and
Wright [1991] or Greenwood and Hercowitz
[1991]). One reason is that home-production models
provide additional margins of adjustment relative
to models without home production. Thus, whereas
in the standard model there are only two uses for
output—consumption and investment—in a homeproduction model there are three—consumption,
investment in market capital, and investment in
nonmarket capital. Likewise, in the standard
model there are only two uses for time—leisure
and work—but in a home-production model there
are three—leisure, market work, and nonmarket
work. In particular, a reduction in return-to-market activity reduces time spent in market work in
the standard model only to the extent that it increases
leisure. In a home-production model, agents can
adjust their market hours by altering the mix of
market work and home activity, even if they do
not change their leisure-time allocation.10
Generalizing the previous model, preferences
are now given by
III. The HomeProduction Model
(10)
Following Becker’s (1965) line of thought, we
now extend the standard model to allow for
nonmarket, or household, production. Some
¥
^ b t [logct + a log(1–nt )] ,
t=0
where ct is an aggregate of market and nonmarket
consumption,
e +(1– m)c e ] 1/e
(11) ct = [mcmt
,
nt
and nt represents the sum of time spent working
in the market and at home,
■ 9 For unmeasured investment, note that along the balanced growth
path, xz = p (g + dz ) z , where all variables have been transformed into
stationary equivalents by dividing the date t by (1+g)t. For the transformed
variables, it follows that
xz
x
p (g + dz ) z
.
= z =
c+x k y–xz k Uk z Uz – p (g + dz ) z
Using the first-order condition for profit maximization, we have
p (i + dz ) = Uz k Uk z Uz –1 ,
where i is the interest rate, and so it follows that xz / (c + xk ) is
independent of p .
■ 10 It should be noted that models with home production cannot
explain observations on market variables that could not be explained, in
principle, by a model without home production (see Benhabib, Rogerson,
and Wright [1991]). That is, one can always choose preferences in a model
without home production that perfectly mimic the market outcomes of a
given model with home production. However, the implied choice of preferences might generally be viewed as nonstandard. For instance, to be consistent with balanced growth and different amounts of market work for different barriers, the implied choice of preferences in a model without home
production would be time dependent; therefore, even if we only want to
look at data on market variables, an advantage of the home production
framework is that it permits one to consider a richer class of specifications
without sacrificing time-independent preferences or balanced growth.
In fact, there are data on how time is allocated across activities, including
home production, and these models provide a structure for interpreting
this data.
(12) nt = nmt + nnt .
The market-production function is unchanged,
t
1–U .
(13) yt = A k U
mt [(1+g) nmt ]
However, a home-production function is now
given by
(14) cnt = A k fnt [(1+g)t nnt ] 1–f .
Exogenous technological change occurs at the
same rate in the home and market sectors, implying that from any initial condition, the model converges to a balanced growth path along which yt ,
cmt , cnt , kmt , and knt all grow at the rate g, while
nmt and nnt remain constant.
29
T A B L E
3
while all home-produced output is consumed.
At this stage, we allow distortionary policies to
differ for the two types of investment goods.
Thus, the two capital stocks evolve according to
Values of p That Will
* = 1/10
Generate y*/ yUS
f» 0
f =0.05
f =0.10
f =0.20
f =0.33
100.0
100.0
100.0
100.0
100.0
e = 0.2
59.3
65.1
71.2
83.7
100.0
e = 0.4
27.6
34.1
42.2
63.1
100.0
e = 0.6
10.3
13.9
19.4
38.9
100.0
e = 0.8
3.3
4.4
6.5
16.4
100.0
e» 0
SOURCE: Authors’ calculations.
T A B L E
4
Values of f and e That Will
* = 1/10
Generate y*/ yUS
f
e
True output ratio
Domestic prices
Welfare cost
U.S. prices
Welfare cost
Steady states
Welfare cost
Dynamics
0.001
0.77
0.42
0.52
1.30
1.18
0.05
0.82
0.43
0.52
1.32
1.21
0.10
0.87
0.44
0.52
1.42
1.24
0.15
0.98
0.45
0.52
1.43
1.25
SOURCE: Authors’ calculations.
An important distinction between the market
and home sectors is that, by assumption, capital
goods can be produced only in the market
sector. That is, market output is still divided
between consumption and investment in the
two types of capital,
(15) cmt + xmt + xnt ² yt ,
■ 11 The other parameters are set as follows: d m = d m = 0.06,
g = 0.02, b = 0.98, and preference parameters are set to generate nm = 0.33
and nn = 0.28 when p = 1.
(16) kmt+1 =(1– dm )kmt +xmt /pm
and
knt+1 =(1– dn )knt +xnt /pn.
We assume that capital is not mobile across
sectors, though this assumption does not
actually affect balanced-growth-path analysis.
The degree to which individuals respond to
economic distortions typically depends on the
other opportunities they face. Home production
is an alternative to market production, and
hence, it may be relevant to evaluating the way
market activity responds to distortions. This
intuition suggests that three parameters are
especially relevant in determining the impact
of the distortions on which we are focusing: 1)
the elasticity of substitution between home and
market consumption, e; 2) the share of capital
in the home sector, f ; and 3) the relative size of
the barriers in the two sectors, pm and pn. For
home production to produce a greater response
of market output to a given investment barrier,
the model requires that individuals be willing to
substitute home consumption for market consumption; such willingness is obviously affected
by e. Additionally, even if individuals are willing
to substitute between market- and homeproduced goods, the distortions on investment
must create an incentive to do so. For this to
occur, either home activity must be less capital
intensive than market activity, or the distortion
to the price of capital used in the nonmarket
sector must be smaller than the distortion to
the price of capital used in the market sector.
We begin with the case in which the barriers
on the two investments are the same, pm = pn= p.
Table 3 displays the value of p that is required to
decrease market output by a factor of 10 relative
to the p = 1 case. Several combinations of values
for the two key parameters, f and e, are considered, and in each case U is set to 1/3.11
As a benchmark, it can be shown formally
that if f = e = 0, the home-production economy
yields predictions for market variables that are
identical to those for the basic model. In other
words, for these parameter settings, home
30
T A B L E
5
Equilibrium Relative to
Undistorted Economy, p m = pn
e =0.0001
e =0.2
e =0.4
e =0.6
e=0.8
y
0.50
0.47
0.41
0.32
0.12
km
0.13
0.12
0.10
0.08
0.03
cm
0.50
0.47
0.41
0.30
0.09
cn
0.90
0.97
1.09
1.31
1.74
x/y
1.00
1.00
1.00
1.00
1.00
y/nm
0.50
0.50
0.50
0.50
0.50
km /nm
0.13
0.13
0.13
0.13
0.13
r
4.00
4.00
4.00
4.00
4.00
nm
1.00
0.94
0.83
0.62
0.22
nn
1.00
1.08
1.21
1.46
1.94
SOURCE: Authors’ calculations.
production does not matter. Hence, the cell
corresponding to U = e » 0 indicates that in the
standard model without home production,
given U = 1/3, we require p = 100 in order to
generate an income differential of 10.
Not surprisingly, the results in table 3 accord
well with the above intuition, in that a higher
elasticity of substitution between the two consumption goods or a lower capital intensity in
■ 12 While it is not clear that one wants to use the same parameter
values in a development context that one uses for studying U.S. business
cycles, we mention here the values of the two key parameters found in
some of the related literature. Estimates of e from micro data in Rupert,
Rogerson, and Wright (1995) and from macro data in McGrattan, Rogerson, and Wright (1997) both yield values close to e = 0.4. Business-cycle
models with home production have a range of values for f; for example, in
Benhabib, Rogerson, and Wright (1992), f = 0.1, while in Greenwood and
Hercowitz (1991) and in McGrattan, Rogerson, and Wright (1997) f is
much bigger. The issue is what one wants to match: lower f generates values for kn /y consistent with measuring home capital in terms of consumer
durables but not residential structures, while higher values are needed to
match kn /y if housing is to be included.
■ 13 The true output ratio is slightly bigger when it is computed
using the price of home-produced output in the undistorted economy
because home-produced goods are relatively more scarce and hence more
expensive in the undistorted economy.
home production implies that a lower barrier is
needed to achieve a given reduction in market
output. However, the quantitative results are
quite striking: if the home technology uses very
little capital and the two consumption goods are
close substitutes, we can reduce the required
barrier from p = 100 in the standard model to
p = 3.3 in our model.12
Table 4 presents similar information in a different way: we fix p = 4, and for various values
of f, we report the value of e needed to yield
y */yUS* = 1/10 (other parameters are as in table 3).
The results confirm our intuition that when
capital’s share in the home is bigger, individuals
must be more willing to substitute between the
two goods to generate the same results. We also
report several other statistics, including the ratio
of total output in the two economies, with total
output computed two ways: market-produced
output plus home-produced output weighted by
its domestic shadow price; and market-produced
output plus home-produced output weighted by
its shadow price in the undistorted economy.
Although the measured market output ratio is
1/10, the true output ratio by either measure is
closer to 1/2, because home production plays a
relatively more important role in the distorted
economy.13 Recall from table 1 that when p = 4
and U = 1/3, the model without home production implies y*/y*US = 1/2. Thus, while the homeproduction model generates much larger ratios
of measured outputs, it generates similar ratios
of true outputs.
Because the true output ratio is so different
from the measured output ratio, it seems interesting to ask about welfare. Table 4 reports the
welfare cost of the barrier in terms of additional
consumption required for an individual who is
indifferent to having or not having the distortion
(for example, a number of 1.25 means that an
agent needs 25 percent more of both market
and home consumption to be as well off with
the barrier as he would be without it). We compute these welfare measures first by simply
comparing steady states—in which case, the
result tells us how much we need to pay an
agent to induce him to not move from a distorted economy to an undistorted economy
already on its balanced growth path—and also
by taking into account transition paths, in which
case the result tells us how much we need to
pay an agent to induce him to not remove the
distortions where he lives. The table reports that
agents must be paid 30 percent– 40 percent of
their consumption to not move, and 18 percent–
25 percent to not remove the barrier. As a
31
T A B L E
6
Equilibrium Relative to
Undistorted Economy, p m > pn
e =0.0001
e =0.2
e =0.4
e =0.6
e =0.8
y
0.50
0.48
0.45
0.40
0.30
km
0.13
0.12
0.11
0.10
0.07
cm
0.50
0.46
0.41
0.31
0.12
cn
0.80
0.83
0.88
0.98
1.17
x /y
1.00
1.08
1.13
1.27
1.72
y/nm
0.50
0.50
0.50
0.50
0.50
km /nm
0.13
0.13
0.13
0.13
0.13
r
4.00
4.00
4.00
4.00
4.00
nm
1.00
0.95
0.90
0.62
0.59
nn
1.00
1.04
1.11
1.46
1.47
SOURCE: Authors’ calculations.
benchmark, in the model without home production, if p = 4, which yields y*/y*US = 1/2, we need
to pay agents 40 percent of their consumption to
not remove the barrier.
Compared to the model without home production, the model with home production generates much larger differences in measured
output, comparable differences in true output,
and smaller welfare differences. What is behind
these results? To answer this question, it is
instructive to look at a larger set of statistics
describing the equilibrium. For f = 0.05, p = 4,
and various values of e, table 5 reports the ratio
of several variables in the distorted and the
undistorted economies (with the other parameters set as above). The first column serves as a
benchmark: with e » 0, the model is very close
to the model without home production, while
■ 14 Also, for larger values of U, the investment share measured
using domestic prices may be increasing in e, since the capital used in the
home sector is produced in the market sector but does not produce measured output. This somewhat reduces the model’s ability to match the fact
discussed above, that there is a relationship between output and savings measured using world prices but not using domestic prices; however,
the magnitude of this effect is relatively small for f less than 0.10 and e
less than 0.60.
the other columns illustrate what happens as
individuals become more willing to substitute
between home and market goods.
Several features are worth noting. First, as e
increases, market activity as measured by y, cm ,
and nm falls, while household activity as measured by cn and n n increases. Second, x / y
is unaffected by the size of p . Hence, the
investment-to-output ratio is not lower in poorer
countries if it is measured using domestic prices,
although it is lower if measured using U.S.
prices (because the U.S. price of capital is 1,
while the domestic price is p ). This matches
well with the data, which shows a relationship
across countries between output and investment,
measured using world prices but not domestic
prices (recall figure 3). Third, although the rental
rate of capital, r, is four times greater in the distorted economy, this is independent of e ; therefore, larger differences in y across countries do
not imply larger differences in r, as is true in the
model without home production. Fourth, while
increases in e reduce market output, they do not
affect the average productivity of labor in the
market: y/nm is half as large in the distorted
economy as it is in the undistorted economy,
independent of e . The key factor behind this
result is that when km falls, so does nm , so that
km /nm stays constant when e increases. Also
notice that as nm decreases, nn increases, so that
total time working remains roughly constant.
The above results are for a relatively low
value of capital’s share in home production,
f = 0.05. As f increases, the overall effect of
adding home production decreases, because
barriers to capital accumulation divert resources
from market to nonmarket activity to a greater
extent when nonmarket activity is relatively less
capital intensive.14 Indeed, if the home and market technologies have the same capital share,
f = U, there is no difference between the models with and without home production in terms
of the barrier required to generate a factor-10
difference in output (see table 3). However, this
is true only if we assume the same barriers to
market and nonmarket capital accumulation,
pm = pn , which is not necessarily the most
interesting case.
Table 6 reports the ratios of variables in one
economy with pm = 4 and pn = 1, and another
economy with pm = pn = 1, assuming equal
capital shares in the home and in the market,
f = U =1/3 (other parameters are set as above).
The distortion affects only market capital accumulation. Even with U, output in the distorted
32
T A B L E
7
Parameters in the Integrated Model
* = 1/10
That Generate y*/ yUS
xz
c+xk
Uz
Uk
e
0
–––
–––
0.42
0.52
1.30
0
0.1
0.25
0.75
0.40
0.49
1.33
0.08
0.2
0.22
0.66
0.30
0.38
1.52
0.18
0.3
0.20
0.50
0.20
0.26
1.93
0.30
0.4
0.19
0.25
0.12
0.16
2.99
0.44
True output ratio True output ratio
Domestic prices
U.S. prices
Welfare cost
Steady states
SOURCE: Authors’ calculations.
economy falls with e, although not by very
much. Asymmetric barriers do not have a significant effect when f = U, even with large values
of e. Although agents want to increase nonmarket activity, when nonmarket activity is capital
intensive, they cannot reduce the size of the
market sector by much because we assume that
household capital must be produced in the
market. While asymmetric barriers imply a
reallocation from market to home production,
the effect will not be significant if household
production is very capital intensive and nonmarket capital can be produced only in the market.
IV. An Integrated
Model
We have discussed how two extended versions
of the standard neoclassical growth model can
account for cross-country income differences,
based on reasonable differences in policies or
institutions that act as barriers to capital accumulation: the model augmented to include intangible capital, and the model augmented to include
household production. These approaches are
not mutually exclusive, however, and in this
section we briefly discuss the implications of
including both intangible capital and household
production in the same model. We will not present the equations explicitly, since it should be
clear how one would combine the two; we simply report the results.
Table 7 presents information for an integrated
structure similar to that of table 4 for the homeproduction model without intangible capital.
Here, we vary intangible capital’s share, Uz , then
choose Uk to match the ratio of measured investment to measured output, and choose e to generate y*/y*US = 1/10 with p = 4. For each Uz , the
table reports Uk , e , and several other statistics.
For the sake of illustration, we set f near zero.
As the importance of intangible capital increases
(larger Uz ), less importance must be assigned to
home production (lower e) to generate y*/y*US =
1/10. Conversely, the more importance one is
willing to assign to household production, the
less one must rely on intangible capital and,
hence, the less unmeasured investment one has
to accept. For example, suppose e =1/2, which is
not far from estimates for the United States. In
this case, Uz = 0.3 generates a ratio of measured
outputs of y*/y*US = 1/10 and implies that unmeasured investment is 30 percent of measured output. It also implies that the ratio of true output is
between 0.20 and 0.26, depending on how
home-produced output is priced. Finally, the
barrier of p = 4 entails a large welfare cost:
an agent would have to receive an additional
93 percent of his consumption to induce him
not to move to an undistorted economy.
V. Evidence
We have demonstrated that, compared to the
model without home production, the model
with home production can generate much larger
differences in measured output, comparable
differences in true output, and smaller differences in welfare. The home-production model
also has some implications that differ from standard models, namely, that hours of market work
will be lower in poor economies. In this section,
we discuss some evidence relating to these
implications.
The prediction that individuals in poorer
economies devote less time to market work is
straightforward; however, it is not easy to test
using conventional data sources, because these
countries do not measure hours of market work
in a systematic fashion. The International Labor
Office publishes statistics on participation rates
for a large number of countries, but this is clearly
different from hours of market work. In fact, participation rates have very little meaning in the
poorest countries, where more than 80 percent
of the population may be engaged in agriculture, much of which is subsistence farming (and
thus better characterized as home rather than
33
market production). Therefore, one must be
somewhat resourceful in evaluating this prediction of the model.
Mueller (1984) uses the Rural Income
Distribution Survey in Botswana. These data
cover agricultural workers and are constructed
from time diaries in which interviewers asked
respondents to account for their time during the
previous day. Interviews took place five times
during the year on various days of the week. The
survey included roughly 4,600 individuals best
described as subsistence farmers. The data present percentages of total time devoted to several
activities, with 12 hours per day as the base.
Time is allocated to each of the following activities: crop husbandry, animal husbandry, wage
labor, trading/vending /processing, hunting/
gathering, repairing/new building, fetching
water, child care, housework, schooling, leisure,
and a few miscellaneous categories. The findings are striking. For males, only about 10 percent
of working time is accounted for by wage labor,
and for females, the figure is closer to 2 percent.
Kirkpatrick (1978) finds similar time allocations
in a survey of the rural sector in Melanesia. In
84 hours per week of daylight, the average adult
in five Melanesian villages spent 19.7 hours on
all phases of agriculture. A good portion of the
remaining waking hours were spent spinning,
weaving, gathering and processing fuels and
food, metalworking, dressing and tanning leather,
manufacturing and repairing tools, and fence
repairing, to name a few. Transportation, recreational, religious, financial, and insurance services are also provided in the household sector.
One measurement issue that must be
addressed is that many poor countries make
some attempt to impute in their GNP accounts
amounts to cover own-use agriculture and, in
some cases, house building. Blades (1975) discusses these issues in detail and finds that in
some countries, the imputed value of subsistence activities is as high as 40 percent of baselevel GNP, although the average for the African
countries in his sample is 10 percent–15 percent.
However, it is important to note that virtually
no attempt is made to impute values for any
services that may be produced in the nonmarket
sector, and it seems likely that this is a sizable
omission. For instance, it seems more than likely
that care for children and the elderly, or financial
and social services, just to name two examples,
are provided to a relatively greater extent outside the formal market in poorer countries.
The poorest countries in the world also have
a great deal of economic activity which could be
classified as illegal or informal, and while this
type of activity may not fit perfectly into the
explicit home-production model analyzed
above, it is in the same spirit. MacGaffey et al.
(1991) report an estimate of total output in Zaire
which includes black-market goods and services
as well as goods and services produced for selfconsumption that is three times larger than output in the official national accounts. They estimate
the size of the black-market economy alone for
other African nations between two-thirds and
one-third of reported output. Important sources
of these estimates are household consumption
surveys. These surveys show that households
consume much more than they earn in wages
and salaries. MacGaffey et al. summarize a
1986 survey for households in Kinshasa, Zaire,
showing that households consumed more than
twice as much as they reportedly earned in
wages and salaries.
Young (1994) documents large increases in
market participation rates for workers in his
study of the economic miracles of East Asia:
Taiwan, Singapore, South Korea, and Hong
Kong (see also Pack [1988]). Over the period
1966–90, the participation rate increased from
0.38 to 0.49 in Hong Kong; 0.27 to 0.51 in Singapore; 0.27 to 0.36 South Korea; and 0.28 to 0.37
in Taiwan. Data for hours per worker are not
reported, but Young claims that only in Hong
Kong have hours per worker declined. Such
increases are consistent with the home-production model. It should be noted, however, that
even in a model in which hours are independent
of distortions along the balanced growth path,
hours will typically change along the transition
from one balanced growth path to another, and
so more time must elapse before these cases
constitute definitive evidence.
One more piece of evidence is contained in
Kuznets (1960) concerning a related but distinct
implication of the analysis in the previous section. In models in which balanced-growth-path
hours of market work are not affected by the
policies under consideration, differences in
income are proportional to differences in average productivity in the market sector. As we
noted following table 5, however, the homeproduction model predicts that differences in the
average productivity of market labor are much
smaller than differences in income. Kuznets presents data for a sample of 33 countries and finds
that differences in the average product of labor
in manufacturing are smaller across rich and
poor countries than are differences in incomes.
In contrast, differences in the average product of
labor in agriculture are greater than differences
in income. One interpretation of this finding that
34
is consistent with our model is that productivity
in agriculture in poor countries is biased downward by systematic overmeasurement of labor
input. This follows from the tendency for all
rural workers to be counted as agricultural
workers, without any attempt to measure hours
devoted to agriculture. If, in fact, substantial
amounts of time are devoted to nonmarket
activities, the average product of labor will be
understated. Mueller (1984), for example,
reports that workers on commercial farms work
more hours per day than workers on noncommercial farms. Livingstone (1981) reports that the
Survey of Rural Workers in Kenya shows rural
workers devote only about half as much time to
agriculture per week than the standard workweek in industry.
In summary, while there are some serious
measurement issues that deserve further study,
and while we would obviously like to have
more and better data, it seems that the information we do have supports explicitly incorporating household production into models of
economic development.
VI. Conclusions
Many economists have suggested that an important difference between rich and poor countries
is the fraction of economic activity in formal
versus informal markets. Most recent work that
tries to account for income differences across
countries abstracts from this feature. In this
paper, we have argued that explicitly incorporating this feature into an otherwise standard model
may enhance the model’s ability to account for
the data. In our model, policies which decrease
the incentive to accumulate capital also lead to a
substitution of economic activity away from the
market sector and into the household-production
sector. Our analysis suggests that this channel
may be quantitatively important.
An implication of our analysis is that poor
countries are relatively not as poor as published
measures of income would indicate. However,
we are not arguing that poor countries are just
like rich countries except that less economic
activity is measured: even when home production plays a big role, we found that poorer
countries are still substantially worse off in terms
of welfare.
To the extent that the output of goods and
services outside formal markets is poorly measured, it is difficult to find direct evidence to support our framework. However, the model has
some predictions that are consistent with empirical findings. For example, Young (1994) reports
that labor participation rates increased substantially in each of the four “Asian tigers” during
their periods of high growth. Also, Kuznets
(1960) reports that international productivity differences are greatest in agricultural and least in
manufacturing, which is consistent with our
approach, if one views measurement of labor
input in manufacturing to be higher quality.
Finally, direct evidence from time diaries in
sub-Saharan Africa supports the finding that
individuals spend a relatively small fraction of
their time in formal market activities. While
more empirical work needs to be done to
corroborate these findings, we conclude that
it may be important to explicitly model the nonmarket sector in the context of studying economic
development.
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